From 2f5106964ad5641581bbd5b01269a38ad02fb5f2 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Mon, 15 Nov 2021 09:25:23 -0800
Subject: [PATCH 01/28] initial commit

---
 demonstrations/tutorial_dqva_mis.py | 380 ++++++++++++++++++++++++++++
 1 file changed, 380 insertions(+)
 create mode 100644 demonstrations/tutorial_dqva_mis.py

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
new file mode 100644
index 0000000000..c03bca6337
--- /dev/null
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -0,0 +1,380 @@
+r""".. _trapped_ions:
+
+Solving MIS with the DQVA Ansatz
+================================
+
+.. meta::
+    :property="og:description": Description and assessment of trapped ion quantum computers
+    :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png
+
+.. related::
+   tutorial_pasqal Quantum computation with neutral atoms
+
+*Author: PennyLane dev team. Posted: 10 November 2021. Last updated: 10 November 2021.*
+
+DQVA ansatz intro
+
+.. container:: alert alert-block alert-info
+    
+    Alert block
+
+"""
+
+##############################################################################
+#
+# QAOA
+# ~~~~~~~~~~~~~~~~~~
+#
+# Something about QAOA
+#
+# .. figure:: ../demonstrations/trapped_ions/confining.png
+#    :align: center
+#    :width: 70%
+#
+#    ..
+#
+#    Confining potential
+#
+# What are the limitations for the general QAOA? 
+#
+# .. figure:: ../demonstrations/trapped_ions/saddle_potential.png
+#    :align: center
+#    :width: 70%
+#
+#    ..
+#
+#    Image caption
+#
+
+
+
+import pennylane as qml
+from pennylane import qaoa
+import numpy as np
+from matplotlib import pyplot as plt
+import networkx as nx
+
+G = nx.Graph()
+G.add_edges_from([(0, 1), (1, 2), (2, 0), (2, 3)])
+
+nx.draw(graph, with_labels=True)
+plt.show()
+
+def hamming_weight(bitstr):
+    return sum([1 for bit in bitstr if bit == '1'])
+    
+def mixer_layer(alpha, init_state, G, mixer_order):
+    """
+    Apply the mixer unitary U_M(alpha) to circ
+
+    Input
+    -----
+    alpha : list[float]
+        The angle values of the parametrized gates
+    init_state : str
+        The current initial state for the ansatz, bits which are "1" are hit
+        with an X-gate at the beginning of the circuit and their partial mixers
+        are turned off. Bitstring is little-endian ordered.
+    G : NetworkX Graph
+        The graph we want to solve MIS on
+    mixer_order : list[int]
+        The order that the partial mixers should be applied in. For a list
+        such as [1,2,0,3] qubit 1's mixer is applied, then qubit 2's, and so on
+    """
+
+    # Apply partial mixers V_i(alpha_i)
+    if mixer_order is None:
+        mixer_order = list(G.nodes)
+    print('Mixer order:', mixer_order)
+
+    # Pad the given alpha parameters to account for the zeroed angles
+    pad_alpha = [None]*len(init_state)
+    next_alpha = 0
+    for qubit in mixer_order:
+        bit = list(reversed(init_state))[qubit]
+        if bit == '1' or next_alpha >= len(alpha):
+            continue
+        else:
+            pad_alpha[qubit] = alpha[next_alpha]
+            next_alpha += 1
+
+    print('init_state: {}, alpha: {}, pad_alpha: {}'.format(init_state,
+                                                              alpha, pad_alpha))
+
+    for qubit in mixer_order:
+        if pad_alpha[qubit] == None or not G.has_node(qubit):
+            # Turn off mixers for qubits which are already 1
+            continue
+
+        neighbors = list(G.neighbors(qubit))
+        anc_idx = 4
+
+        qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors))
+
+        # apply an X rotation controlled by the state of the ancilla qubit
+        qml.CRX(2*pad_alpha[qubit], wires = [anc_idx, qubit])
+        
+        qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors))
+
+def cost_layer(gamma, G):
+    """
+    Apply a parameterized Z-rotation to every qubit
+    """
+    for qb in G.nodes:
+        qml.RZ(2*gamma, wires=qb)
+
+def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None):
+    nq = len(G.nodes)
+    print(nq)
+
+    # Step 1: Jump Start
+    # Run an efficient classical approximation algorithm to warm-start the optimization
+    if init_state is None:
+        init_state = '0'*nq
+
+    # Step 2: Mixer Initialization
+    for qb, bit in enumerate(reversed(init_state)):
+        if bit == '1':
+            qml.PauliX(wires = qb)
+            
+    # parse the variational parameters
+    # The dqva ansatz dynamically turns off partial mixers for qubits in |1>
+    # and adds extra mixers to the end of the circuit
+    num_nonzero = nq - hamming_weight(init_state)
+    assert (len(params) == (nq + 1) * P), "Incorrect number of parameters!"
+    alpha_list = []
+    gamma_list = []
+    last_idx = 0
+    for p in range(P):
+        chunk = num_nonzero + 1
+        cur_section = params[p*chunk:(p+1)*chunk]
+        alpha_list.append(cur_section[:-1])
+        gamma_list.append(cur_section[-1])
+        last_idx = (p+1)*chunk
+
+    # Add the leftover parameters as extra mixers
+    alpha_list.append(params[last_idx:])
+
+    for i in range(len(alpha_list)):
+        print('alpha_{}: {}'.format(i, alpha_list[i]))
+        if i < len(gamma_list):
+            print('gamma_{}: {}'.format(i, gamma_list[i]))
+
+    # Construct the dqva ansatz
+    #for alphas, gamma in zip(alpha_list, gamma_list):
+    for i in range(len(alpha_list)):
+        alphas = alpha_list[i]
+        mixer_layer(alphas, init_state, G,  mixer_order)
+
+
+        if i < len(gamma_list):
+            gamma = gamma_list[i]
+            cost_layer(gamma, G)
+##############################################################################
+# Text in between code goes here
+
+dev = qml.device("default.qubit", wires=1)
+
+
+@qml.qnode(dev)
+def ion_hadamard(state):
+
+    if state == 1:
+        qml.PauliX(wires=0)
+    
+    """We use a series of seemingly arbitrary pulses that will give the Hadamard gate.
+    Why this is the case will become clear later"""
+
+    qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0)
+    qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0)
+    qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0)
+    qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0)
+    qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0)
+
+    return qml.state()
+
+#For comparison, we use the Hadamard built into PennyLane
+@qml.qnode(dev)
+def hadamard(state):
+
+    if state == 1:
+        qml.PauliX(wires=0)
+
+    qml.Hadamard(wires=0)
+
+    return qml.state()
+
+#We confirm that the values given by both functions are the same up to numerical error
+print(np.isclose(1j * ion_hadamard(0), hadamard(0)))
+print(np.isclose(1j * ion_hadamard(1), hadamard(1)))
+
+##############################################################################
+# Note that the desired gate was obtained up to a global phase factor.
+# A similar exercise can be done for the :math:`T` gate:
+
+
+@qml.qnode(dev)
+def ion_Tgate(state):
+
+    if state == 1:
+        qml.PauliX(wires=0)
+
+    qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0)
+    qml.QubitUnitary(evolution(np.pi / 2, np.pi / 4 / Omega), wires=0)
+    qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0)
+
+    return qml.state()
+
+
+@qml.qnode(dev)
+def tgate(state):
+
+    if state == 1:
+        qml.PauliX(wires=0)
+
+    qml.T(wires=0)
+
+    return qml.state()
+
+
+print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(0), tgate(0)))
+print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(1), tgate(1)))
+
+##############################################################################
+# Text
+
+import matplotlib.pyplot as plt
+
+
+@qml.qnode(dev)
+def evolution_prob(t):
+
+    qml.QubitUnitary(evolution(0, t / Omega), wires=0)
+
+    return qml.probs(wires=0)
+
+
+t = np.linspace(0, 4 * np.pi, 101)
+s = [evolution_prob(i)[1].numpy() for i in t]
+
+fig1, ax1 = plt.subplots(figsize=(9, 6))
+
+ax1.plot(t, s, color="#9D2EC5")
+
+ax1.set(
+    xlabel="time (in units of 1/Ω)", 
+    ylabel="Probability", 
+    title="Probability of measuring the excited state"
+)
+ax1.grid()
+
+plt.show()
+
+
+#
+# References
+# ----------
+#
+# .. [#DiVincenzo2000]
+#
+#     D. DiVincenzo. (2000) "The Physical Implementation of Quantum Computation",
+#     `Fortschritte der Physik 48 (9–11): 771–783
+#     <https://onlinelibrary.wiley.com/doi/10.1002/1521-3978(200009)48:9/11%3C771::AID-PROP771%3E3.0.CO;2-E>`__.
+#     (`arXiv <https://arxiv.org/abs/quant-ph/0002077>`__)
+#
+# .. [#Paul1953]
+#
+#     W. Paul, H. Steinwedel. (1953) "Ein neues Massenspektrometer ohne Magnetfeld",
+#     RZeitschrift für Naturforschung A 8 (7): 448-450.
+#
+# .. [#CiracZoller]
+#
+#     J. Cirac, P. Zoller. (1995) "Quantum Computations with Cold Trapped Ions".
+#     Physical Review Letters 74 (20): 4091–4094.
+#
+# .. [#Malinowski]
+#
+#     M. Malinowski. (2021) "Unitary and Dissipative Trapped-​Ion Entanglement Using
+#     Integrated Optics". PhD Thesis retrieved from `ETH thesis repository
+#     <https://ethz.ch/content/dam/ethz/special-interest/phys/quantum-electronics/tiqi-dam/documents/phd_theses/Thesis-Maciej-Malinowski>`__.
+#
+# .. [#NandC2000]
+#
+#     M. A. Nielsen, and I. L. Chuang (2000) "Quantum Computation and Quantum Information",
+#     Cambridge University Press.
+#
+# .. [#Hughes2020]
+#
+#     A. Hughes, V. Schafer, K. Thirumalai, et al. (2020)
+#     "Benchmarking a High-Fidelity Mixed-Species Entangling Gate"
+#     `Phys. Rev. Lett. 125, 080504
+#     <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.080504>`__.
+#     (`arXiv <https://arxiv.org/abs/2004.08162>`__)
+#
+# .. [#Bergou2021]
+#
+#     J. Bergou, M. Hillery, and M. Saffman. (2021) "Quantum Information Processing",
+#     Springer.
+#
+# .. [#Molmer1999]
+#
+#     A. Sørensen, K. Mølmer.  (1999) "Multi-particle entanglement of hot trapped ions",
+#     `Physical Review Letters. 82 (9): 1835–1838
+#     <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.1835>`__.
+#     (`arXiv <https://arxiv.org/abs/quant-ph/9810040>`__)
+#
+# .. [#Brown2019]
+#
+#     M. Brown, M. Newman, and K. Brown. (2019)
+#     "Handling leakage with subsystem codes",
+#     `New J. Phys. 21 073055
+#     <https://iopscience.iop.org/article/10.1088/1367-2630/ab3372>`__.
+#     (`arXiv <https://arxiv.org/abs/1903.03937>`__)
+#
+# .. [#Monroe2014]
+#
+#     C. Monroe, R. Ruassendorf, A Ruthven, et al. (2019)
+#     "Large scale modular quantum computer architecture with atomic memory and photonic interconnects",
+#     `Phys. Rev. A 89 022317
+#     <https://journals.aps.org/pra/abstract/10.1103/PhysRevA.89.022317>`__.
+#     (`arXiv <https://arxiv.org/abs/1208.0391>`__)
+#
+# .. [#QCCD2002]
+#
+#     D. Kielpinski, C. Monroe, and D. Wineland. (2002)
+#     "Architecture for a large-scale ion-trap quantum computer",
+#     `Nature 417, 709–711 (2002).
+#     <https://www.nature.com/articles/nature00784>`__.
+#
+# .. [#Amini2010]
+#
+#     J. Amini, H. Uys, J. Wesenberg, et al. (2010)
+#     "Toward scalable ion traps for quantum information processing",
+#     `New J. Phys 12 033031
+#     <https://iopscience.iop.org/article/10.1088/1367-2630/12/3/033031/meta>`__.
+#     (`arXiv <https://arxiv.org/abs/0909.2464>`__)
+#
+#
+# .. [#Pino2021]
+#
+#     J. Pino, J. Dreiling, J, C, Figgatt, et al. (2021)
+#     "Demonstration of the trapped-ion quantum CCD computer architecture".
+#     `Nature 592, 209–213
+#     <https://www.nature.com/articles/s41586-021-03318-4>`__.
+#     (`arXiv <https://arxiv.org/abs/2003.01293>`__)
+#
+# .. [#Blumel2021]
+#
+#     R. Blumel, N. Grzesiak, N. Nguyen, et al. (2021)
+#     "Efficient Stabilized Two-Qubit Gates on a Trapped-Ion Quantum Computer"
+#     `Phys. Rev. Lett. 126, 220503
+#     <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.220503>`__.
+#     (`arXiv <https://arxiv.org/abs/2101.07887>`__)
+#
+# .. [#Niffenegger2020]
+#
+#     R. Niffenegger, J. Stuart, C.Sorace-Agaskar, et al. (2020)
+#     "Integrated multi-wavelength control of an ion qubit"
+#     `Nature volume 586, pages538–542
+#     <https://www.nature.com/articles/s41586-020-2811-x>`__.
+#     (`arXiv <https://arxiv.org/abs/2001.05052>`__)

From 7cce680e9bcb2c62c8aa48ff01bc03769267dfd9 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Fri, 19 Nov 2021 18:19:41 -0800
Subject: [PATCH 02/28] init

---
 demonstrations/tutorial_dqva_mis.py | 213 +---------------------------
 1 file changed, 6 insertions(+), 207 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index c03bca6337..0408cf2594 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -10,7 +10,7 @@
 .. related::
    tutorial_pasqal Quantum computation with neutral atoms
 
-*Author: PennyLane dev team. Posted: 10 November 2021. Last updated: 10 November 2021.*
+*Author: *
 
 DQVA ansatz intro
 
@@ -33,17 +33,10 @@
 #
 #    ..
 #
-#    Confining potential
+#    Image caption
 #
 # What are the limitations for the general QAOA? 
 #
-# .. figure:: ../demonstrations/trapped_ions/saddle_potential.png
-#    :align: center
-#    :width: 70%
-#
-#    ..
-#
-#    Image caption
 #
 
 
@@ -57,7 +50,7 @@
 G = nx.Graph()
 G.add_edges_from([(0, 1), (1, 2), (2, 0), (2, 3)])
 
-nx.draw(graph, with_labels=True)
+nx.draw(G, with_labels=True)
 plt.show()
 
 def hamming_weight(bitstr):
@@ -173,208 +166,14 @@ def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None):
 ##############################################################################
 # Text in between code goes here
 
-dev = qml.device("default.qubit", wires=1)
-
-
-@qml.qnode(dev)
-def ion_hadamard(state):
-
-    if state == 1:
-        qml.PauliX(wires=0)
-    
-    """We use a series of seemingly arbitrary pulses that will give the Hadamard gate.
-    Why this is the case will become clear later"""
-
-    qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0)
-    qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0)
-    qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0)
-    qml.QubitUnitary(evolution(np.pi / 2, np.pi / 2 / Omega), wires=0)
-    qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0)
-
-    return qml.state()
-
-#For comparison, we use the Hadamard built into PennyLane
-@qml.qnode(dev)
-def hadamard(state):
-
-    if state == 1:
-        qml.PauliX(wires=0)
-
-    qml.Hadamard(wires=0)
-
-    return qml.state()
-
-#We confirm that the values given by both functions are the same up to numerical error
-print(np.isclose(1j * ion_hadamard(0), hadamard(0)))
-print(np.isclose(1j * ion_hadamard(1), hadamard(1)))
-
-##############################################################################
-# Note that the desired gate was obtained up to a global phase factor.
-# A similar exercise can be done for the :math:`T` gate:
-
-
-@qml.qnode(dev)
-def ion_Tgate(state):
-
-    if state == 1:
-        qml.PauliX(wires=0)
-
-    qml.QubitUnitary(evolution(0, -np.pi / 2 / Omega), wires=0)
-    qml.QubitUnitary(evolution(np.pi / 2, np.pi / 4 / Omega), wires=0)
-    qml.QubitUnitary(evolution(0, np.pi / 2 / Omega), wires=0)
-
-    return qml.state()
-
-
-@qml.qnode(dev)
-def tgate(state):
-
-    if state == 1:
-        qml.PauliX(wires=0)
-
-    qml.T(wires=0)
-
-    return qml.state()
-
-
-print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(0), tgate(0)))
-print(np.isclose(np.exp(1j * np.pi / 8) * ion_Tgate(1), tgate(1)))
-
-##############################################################################
-# Text
-
-import matplotlib.pyplot as plt
-
-
-@qml.qnode(dev)
-def evolution_prob(t):
-
-    qml.QubitUnitary(evolution(0, t / Omega), wires=0)
-
-    return qml.probs(wires=0)
-
-
-t = np.linspace(0, 4 * np.pi, 101)
-s = [evolution_prob(i)[1].numpy() for i in t]
-
-fig1, ax1 = plt.subplots(figsize=(9, 6))
-
-ax1.plot(t, s, color="#9D2EC5")
-
-ax1.set(
-    xlabel="time (in units of 1/Ω)", 
-    ylabel="Probability", 
-    title="Probability of measuring the excited state"
-)
-ax1.grid()
-
-plt.show()
 
 
 #
 # References
 # ----------
 #
-# .. [#DiVincenzo2000]
-#
-#     D. DiVincenzo. (2000) "The Physical Implementation of Quantum Computation",
-#     `Fortschritte der Physik 48 (9–11): 771–783
-#     <https://onlinelibrary.wiley.com/doi/10.1002/1521-3978(200009)48:9/11%3C771::AID-PROP771%3E3.0.CO;2-E>`__.
-#     (`arXiv <https://arxiv.org/abs/quant-ph/0002077>`__)
-#
-# .. [#Paul1953]
-#
-#     W. Paul, H. Steinwedel. (1953) "Ein neues Massenspektrometer ohne Magnetfeld",
-#     RZeitschrift für Naturforschung A 8 (7): 448-450.
-#
-# .. [#CiracZoller]
-#
-#     J. Cirac, P. Zoller. (1995) "Quantum Computations with Cold Trapped Ions".
-#     Physical Review Letters 74 (20): 4091–4094.
-#
-# .. [#Malinowski]
-#
-#     M. Malinowski. (2021) "Unitary and Dissipative Trapped-​Ion Entanglement Using
-#     Integrated Optics". PhD Thesis retrieved from `ETH thesis repository
-#     <https://ethz.ch/content/dam/ethz/special-interest/phys/quantum-electronics/tiqi-dam/documents/phd_theses/Thesis-Maciej-Malinowski>`__.
-#
-# .. [#NandC2000]
-#
-#     M. A. Nielsen, and I. L. Chuang (2000) "Quantum Computation and Quantum Information",
-#     Cambridge University Press.
-#
-# .. [#Hughes2020]
-#
-#     A. Hughes, V. Schafer, K. Thirumalai, et al. (2020)
-#     "Benchmarking a High-Fidelity Mixed-Species Entangling Gate"
-#     `Phys. Rev. Lett. 125, 080504
-#     <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.080504>`__.
-#     (`arXiv <https://arxiv.org/abs/2004.08162>`__)
-#
-# .. [#Bergou2021]
-#
-#     J. Bergou, M. Hillery, and M. Saffman. (2021) "Quantum Information Processing",
-#     Springer.
-#
-# .. [#Molmer1999]
-#
-#     A. Sørensen, K. Mølmer.  (1999) "Multi-particle entanglement of hot trapped ions",
-#     `Physical Review Letters. 82 (9): 1835–1838
-#     <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.1835>`__.
-#     (`arXiv <https://arxiv.org/abs/quant-ph/9810040>`__)
-#
-# .. [#Brown2019]
-#
-#     M. Brown, M. Newman, and K. Brown. (2019)
-#     "Handling leakage with subsystem codes",
-#     `New J. Phys. 21 073055
-#     <https://iopscience.iop.org/article/10.1088/1367-2630/ab3372>`__.
-#     (`arXiv <https://arxiv.org/abs/1903.03937>`__)
-#
-# .. [#Monroe2014]
-#
-#     C. Monroe, R. Ruassendorf, A Ruthven, et al. (2019)
-#     "Large scale modular quantum computer architecture with atomic memory and photonic interconnects",
-#     `Phys. Rev. A 89 022317
-#     <https://journals.aps.org/pra/abstract/10.1103/PhysRevA.89.022317>`__.
-#     (`arXiv <https://arxiv.org/abs/1208.0391>`__)
-#
-# .. [#QCCD2002]
-#
-#     D. Kielpinski, C. Monroe, and D. Wineland. (2002)
-#     "Architecture for a large-scale ion-trap quantum computer",
-#     `Nature 417, 709–711 (2002).
-#     <https://www.nature.com/articles/nature00784>`__.
-#
-# .. [#Amini2010]
-#
-#     J. Amini, H. Uys, J. Wesenberg, et al. (2010)
-#     "Toward scalable ion traps for quantum information processing",
-#     `New J. Phys 12 033031
-#     <https://iopscience.iop.org/article/10.1088/1367-2630/12/3/033031/meta>`__.
-#     (`arXiv <https://arxiv.org/abs/0909.2464>`__)
-#
-#
-# .. [#Pino2021]
-#
-#     J. Pino, J. Dreiling, J, C, Figgatt, et al. (2021)
-#     "Demonstration of the trapped-ion quantum CCD computer architecture".
-#     `Nature 592, 209–213
-#     <https://www.nature.com/articles/s41586-021-03318-4>`__.
-#     (`arXiv <https://arxiv.org/abs/2003.01293>`__)
-#
-# .. [#Blumel2021]
-#
-#     R. Blumel, N. Grzesiak, N. Nguyen, et al. (2021)
-#     "Efficient Stabilized Two-Qubit Gates on a Trapped-Ion Quantum Computer"
-#     `Phys. Rev. Lett. 126, 220503
-#     <https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.220503>`__.
-#     (`arXiv <https://arxiv.org/abs/2101.07887>`__)
+# .. [#Saleem2020]
 #
-# .. [#Niffenegger2020]
+#     Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization",
+#     `arXiv preprint arXiv:2010.06660 <https://arxiv.org/abs/2010.06660>`__.
 #
-#     R. Niffenegger, J. Stuart, C.Sorace-Agaskar, et al. (2020)
-#     "Integrated multi-wavelength control of an ion qubit"
-#     `Nature volume 586, pages538–542
-#     <https://www.nature.com/articles/s41586-020-2811-x>`__.
-#     (`arXiv <https://arxiv.org/abs/2001.05052>`__)

From 9729175c0909aa55c2f182de6e2fabaa68abe776 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Fri, 14 Jan 2022 22:19:28 -0800
Subject: [PATCH 03/28] add demo

---
 demonstrations/dqva_mis/independent-sets.png | Bin 0 -> 113218 bytes
 demonstrations/dqva_mis/mixer-unitary.png    | Bin 0 -> 63942 bytes
 demonstrations/dqva_mis/partial-mixers.png   | Bin 0 -> 160156 bytes
 demonstrations/tutorial_dqva_mis.py          | 798 ++++++++++++++++---
 test-results/sphinx-gallery/junit.xml        |   1 +
 5 files changed, 695 insertions(+), 104 deletions(-)
 create mode 100644 demonstrations/dqva_mis/independent-sets.png
 create mode 100644 demonstrations/dqva_mis/mixer-unitary.png
 create mode 100644 demonstrations/dqva_mis/partial-mixers.png
 create mode 100644 test-results/sphinx-gallery/junit.xml

diff --git a/demonstrations/dqva_mis/independent-sets.png b/demonstrations/dqva_mis/independent-sets.png
new file mode 100644
index 0000000000000000000000000000000000000000..ad5457de130e700d1b703efe398add02345a7255
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HcmV?d00001

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 0408cf2594..9dba349f1b 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -1,140 +1,569 @@
-r""".. _trapped_ions:
+r""".. _dqva_mis:
 
-Solving MIS with the DQVA Ansatz
-================================
+Solving MIS with the QAO-Ansatz and the DQVA
+============================================
 
 .. meta::
     :property="og:description": Description and assessment of trapped ion quantum computers
     :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png
 
 .. related::
-   tutorial_pasqal Quantum computation with neutral atoms
+   tutorial_qaoa_intro Introduction to QAOA
+   tutorial_qaoa_maxcut QAOA for MaxCut
 
-*Author: *
-
-DQVA ansatz intro
+*Author: Priya Angara*
 
+This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) <https://arxiv.org/abs/2010.06660>` and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) <https://arxiv.org/abs/1709.03489)>` for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. 
 .. container:: alert alert-block alert-info
     
     Alert block
 
 """
 
-##############################################################################
-#
-# QAOA
-# ~~~~~~~~~~~~~~~~~~
-#
-# Something about QAOA
-#
-# .. figure:: ../demonstrations/trapped_ions/confining.png
+######################################################################
+# QAOA, QAO-Ansatz, and DQVA
+# --------------------------
+# 
+# The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid
+# quantum-classical algorithm by `Farhi et
+# al. <https://arxiv.org/abs/1411.4028>`__ is a significant meta-heuristic
+# for finding solutions to combinatorial optimization problems. The
+# quantum part involves alternating between a *cost Hamiltonian*,
+# :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evalulate the
+# objective function and a classical optimization loop updates the ansatz
+# parameters. You can learn more about this in `PennyLane's tutorial on
+# QAOA <https://pennylane.ai/qml/demos/tutorial_qaoa_intro.html>`__.
+# 
+# For solving constrained combinatorial optimization problems such as MIS
+# using QAOA, constraints can be imposed in two ways: 1. By adding a
+# penalty term to the problem's objective function (the Lagrange
+# multiplier approach) 2. By constructing the variational ansatz in a way
+# such that constraints are satisfied at all times
+# 
+# Hadfield et al., extend the QAOA framework and introduce the **Quantum
+# Alternating Operator Ansatz (QAO-Ansatz)** which alternates between more
+# general families of operators thereby supporting potentially more useful
+# sets of states by focusing on unitaries rather than Hamiltonians. For
+# example, a mixing operator :math:`U_M (\beta)` is decomposed into a
+# sequence of partial mixers :math:`U_{M, x}` which may not correspond to
+# time evolution under a fixed mixer Hamiltonian :math:`H_M`.
+# 
+# .. figure:: ../demonstrations/dqva_mis/independent_sets.png
 #    :align: center
-#    :width: 70%
+#    :width: 100%
 #
 #    ..
 #
-#    Image caption
-#
-# What are the limitations for the general QAOA? 
+#    Independent sets of a graph
+# 
+# The QAO-Ansatz guarantees that the output never leaves the set of
+# feasible states (in contrast, the penalty term optimization requires an
+# additional pruning step since the output contains both feasible and
+# infeasible solutions). However, the QAO-Ansatz requires more complex
+# quantum circuitry, which in the case of the MIS, manifests itself as
+# Multi-Controlled Toffoli gates that require high connectivity between
+# qubits. This makes the QAO-Ansatz inadequate for large graphs on current
+# near-term quantum computers.
+# 
+# This brings us to the **Dynamic Quantum Variational Ansatz (DQVA)** that
+# maximizes the performance of the QAO-Ansatz by: 
+# 
+# 1. *Warm starting the optimization* by starting with an initial state that is a feasible state
+# or a superposition of feasible states. A feasible state can be found by
+# using a classical approximate polynomial time algorithm. 
+# 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* 
+# 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries
+# 
+# Now that we have the basics in place, let's implement the QAO-Ansatz and
+# the DQVA!
+
+# 
+
+
+######################################################################
+# Formulating the Maximum Independent Set (MIS) problem
+# -----------------------------------------------------
+# 
+# Given a graph, :math:`G = (V, E)` where :math:`V` is the set of vertices
+# and :math:`E` is the set of edges, an *independent set* is a subset of
+# vertices, :math:`V' \subset V`, such that no two vertices in :math:`V'`
+# share an edge. A *Maximum Independet Set* (MIS) is an independent set of
+# the largest possible size. In this demo, our objective is to find the
+# MIS of a given graph. The following image shows some independent sets of
+# a graph with 5 vertices:
+# 
+# .. figure:: ../demonstrations/dqva_mis/independent_sets.png
+#    :align: center
+#    :width: 100%
 #
+#    ..
 #
-
-
+#    Independent sets of a graph
+# 
+# Here, independent sets can be represented as bitstrings
+# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown
+# above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are
+# independent sets of size 0, 1, and 2 respectively. The Minimum Vertex
+# Cover, i.e, vertices that cover all edges, is the dual of MIS.
+# 
+# We will start with the QAO-Ansatz and then re-use most components to
+# formulate the DQVA.
+# 
+# First, we start with the necessary imports.
+# 
 
 import pennylane as qml
 from pennylane import qaoa
-import numpy as np
-from matplotlib import pyplot as plt
+from pennylane import numpy as np
 import networkx as nx
+from matplotlib import pyplot as plt
+import copy
+
 
-G = nx.Graph()
-G.add_edges_from([(0, 1), (1, 2), (2, 0), (2, 3)])
+######################################################################
+# Next, we define the graph shown above with 5 nodes for which we would
+# like to find the Maximum Independent Set.
+# 
 
-nx.draw(G, with_labels=True)
+edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)]
+# edges = [(0, 1), (0, 3), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)]
+graph = nx.Graph(edges)
+nx.draw(graph, with_labels=True)
 plt.show()
 
+
+######################################################################
+# We will also instantiate a device with the number of wires being one
+# more than the number of nodes (the extra wire is an ancilla).
+# 
+
+wires = len(graph.nodes) + 1
+dev = qml.device("default.qubit", wires=range(wires))
+
+
+######################################################################
+# For MIS, our goal is to maximize the number of vertices that form an
+# independent set, which is equivalent to maximizing the `Hamming
+# weight <https://en.wikipedia.org/wiki/Hamming_weight>`__ of the
+# bitstring. Therefore, our objective function is the Hamming weight
+# operator,
+# 
+# .. math::
+# 
+# 
+#    C_{obj} = H = \sum_{i \in V} b_i,
+# 
+# where
+# 
+# .. math::
+# 
+# 
+#    b_i = \frac{1}{2} (1 - Z_i).
+# 
+# Here :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th
+# qubit.
+# 
+
+
+######################################################################
+# We need a couple of helper functions here: ``hamming_weight``, which
+# calculates the Hamming weight of a given bitstring and ``is_indset``,
+# which verifies whether a given bitstring is a valid independent set.
+# 
+
 def hamming_weight(bitstr):
     return sum([1 for bit in bitstr if bit == '1'])
+
+def is_indset(bitstr, G):
+    nodes = list(G.nodes)
+    ind_set = []
+    for idx, bit in enumerate(bitstr):
+        if bit == '1':
+            cur_neighbors = list(G.neighbors(idx))
+            for node in ind_set:
+                if node in cur_neighbors:
+                    return False
+            else:
+                ind_set.append(idx)
+    return True
+
+
+######################################################################
+# Next, we define a cost unitary that incorporates the objective function,
+# :math:`H` and is parameterized by :math:`\gamma`:
+# 
+# .. math::
+# 
+# 
+#    U_C(\gamma) = \prod_i e^{i \gamma H}. 
+# 
+
+def cost_layer(gamma):
+    for qb in graph.nodes:
+        qml.RZ(2*gamma, wires=qb)
+
+
+######################################################################
+# The mixer unitary for the QAO-Ansatz is defined as $
+# U_M(:raw-latex:`\beta`) = :raw-latex:`\prod`\_i
+# e^{i:raw-latex:`\beta `M_i}$
+# 
+# where
+# 
+# .. math::
+# 
+# 
+#    M_i = X_i \tilde{B},
+# 
+# and
+# 
+# .. math::
+#
+# 
+#    \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}.
+# 
+# Here, where :math:`v_j` are the neighbors and :math:`l` is the number of
+# neighbors for the :math:`i`-th node and
+# 
+# .. math::
+# 
+# 
+#    \tilde{b}_{v_j} = \frac{1+Z_{v_j}}{2}.
+# 
+# The mixer unitary can also be written as a product of :math:`N` partial
+# mixers :math:`V_i`:
+# 
+# .. math::
+# 
+# 
+#    U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N  (I + (e^{-i\beta X_i} - I)\tilde{B}) $
+# 
+# The partial mixers may not commute with each other, i.e.,
+# :math:`[V_i, V_j] \neq 0` therefore, different orderings of these mixers
+# may have different outputs. Therefore, the variational ansatz is defined
+# up to a permutation:
+# 
+# .. math::
+# 
+# 
+#    U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_N(\beta))
+# 
+# where P is the permutation's function of labels from :math:`1` to
+# :math:`N`.
+# 
+# These partial mixers are implemented using parameterized
+# controlled-:math:`X` rotations sandwiched between two `multi-controlled
+# Toffoli
+# gates <https://pennylane.readthedocs.io/en/stable/code/api/pennylane.MultiControlledX.html>`__.
+# The number of qubits we require is one more than the number of vertices.
+# This extra qubit is used as an ancilliary qubit to serve as the target
+# for the multi-controlled Toffoli gates.
+# 
+# .. figure:: ../demonstrations/dqva_mis/mixer_unitary.png
+#    :align: center
+#    :width: 100%
+#
+#    ..
+#
+#    Formulation of the mixer unitary
+# 
+
+def mixer_layer(beta, ancilla, mixer_order):
     
-def mixer_layer(alpha, init_state, G, mixer_order):
-    """
-    Apply the mixer unitary U_M(alpha) to circ
-
-    Input
-    -----
-    alpha : list[float]
-        The angle values of the parametrized gates
-    init_state : str
-        The current initial state for the ansatz, bits which are "1" are hit
-        with an X-gate at the beginning of the circuit and their partial mixers
-        are turned off. Bitstring is little-endian ordered.
-    G : NetworkX Graph
-        The graph we want to solve MIS on
-    mixer_order : list[int]
-        The order that the partial mixers should be applied in. For a list
-        such as [1,2,0,3] qubit 1's mixer is applied, then qubit 2's, and so on
-    """
-
-    # Apply partial mixers V_i(alpha_i)
+    # Permute the order of mixing unitaries
     if mixer_order is None:
-        mixer_order = list(G.nodes)
-    print('Mixer order:', mixer_order)
+        mixer_order = list(graph.nodes)
+    
+    for qubit in list(mixer_order):
+        neighbors = list(graph.neighbors(qubit))
+
+        # Apply the multi-controlled Toffoli, targetting the ancilla qubit
+        ctrl_qubits = [i for i in neighbors]
+        qml.MultiControlledX(control_wires = ctrl_qubits,
+                             wires = [ancilla], 
+                             control_values ='0'*len(neighbors))
+        
+        qml.CRX(2*beta, wires=[ancilla, qubit])
+        
+        # Uncompute the ancilla
+        qml.MultiControlledX(control_wires = ctrl_qubits,
+                             wires = [ancilla], 
+                             control_values ='0'*len(neighbors))
+
+
+######################################################################
+# We are now ready to build the circuit consisting of alternating cost and
+# mixer layers. A list of ``params`` includes parameters for the mixer
+# layer (even elements) and cost layer (odd elements).
+# 
 
-    # Pad the given alpha parameters to account for the zeroed angles
+def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
+    nq = len(graph.nodes)
+    # Initialize
+    if init_state is None:
+        init_state = '0'*nq
+        
+    else:
+        for qb, bit in enumerate(reversed(init_state)):
+            if bit == '1':
+                qml.PauliX(wires=qb)
+    assert (len(params) == 2*P),"Incorrect number of parameters!"
+    
+    betas  = [a for i, a in enumerate(params) if i % 2 == 0]
+    gammas = [a for i, a in enumerate(params) if i % 2 == 1]
+    
+    for beta, gamma in zip(betas, gammas):
+        ancilla = nq # Use the last qubit as an ancilla
+        mixer_layer(beta, ancilla, mixer_order)
+        cost_layer(gamma)
+
+
+
+######################################################################
+# Now, let's build the circuit that finds the probabilities of measuring
+# each bitstring.
+# 
+
+@qml.qnode(dev)
+def probability_circuit(P, params=[], init_state=None, mixer_order=None):
+    qaoa_ansatz(P, params, init_state, mixer_order)
+    return qml.probs(wires=range(wires-1))
+
+
+######################################################################
+# Finally, we run the quantum-classical loop for ``m`` different mixer
+# permutations. The expectation value is also calculated in the cost
+# function ``f`` by running the probability circuit and calculating the
+# Hamming weight (which is defined as the cost). In each iteration, an
+# ansatz is constructed based on the mixer order, with optimal parameters
+# from classical minimization of the cost function (Gradient Descent)
+#
+# .. math::
+# 
+# 
+#    \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle. 
+# 
+
+def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None,  threshold=1e-5, cutoff=1):
+    # Select an ordering for the partial mixers
+    if mixer_order == None:
+        cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
+    else:
+        cur_permutation = mixer_order
+
+    # Define the function to be optimized
+    # Note that we are returning -cost since this is a minimization
+    def f(params):
+        probs = probability_circuit(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation)
+        avg_cost = 0
+        for sample in range(0, len(probs)):
+            
+            x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))]
+            # Cost function is Hamming weight
+            avg_cost += probs[sample] * sum(x)
+        
+        return -avg_cost
+    
+    # Begin outer optimization loop
+    best_indset = init_state
+    best_init_state = init_state
+    cur_init_state = init_state
+    best_params = None
+    best_perm = copy.copy(cur_permutation)
+    
+    # Randomly permute the order of mixer unitaries m times
+    for mixer_round in range (1, m+1):
+     
+        new_hamming_weight = hamming_weight(cur_init_state)
+        inner_round = 1
+         
+        while (inner_round < 2):
+            print('Start round {}.{}, Initial state = {}'.format(mixer_round, inner_round, cur_init_state))
+
+            # Begin inner variational loop
+            num_params = 2 * P
+            print('\tNum params =', num_params)
+
+            init_params = np.random.uniform(low=-np.pi, high=np.pi, size=num_params)
+            print('\tCurrent Mixer Order:', cur_permutation)
+
+            # Optimize parameters
+            optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
+            cur_params = init_params.copy()
+            
+            for i in range(70):
+                cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
+
+            opt_params = cur_params.copy()
+
+            print('\tOptimal cost:', opt_cost)
+
+            # Obtain probabilites
+            probs = probability_circuit(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation)
+            top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1]
+            # print (top_counts)
+            
+            best_hamming_weight = hamming_weight(best_indset)
+            better_strs = []
+            
+            print(top_counts[:cutoff])
+            for bitstr in top_counts[:cutoff]:
+                
+                this_hamming = hamming_weight(bitstr)
+                if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
+                    better_strs.append((bitstr, this_hamming))
+                    
+            better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
+
+            # If no improvement was made, break and go to next mixer round
+            if len(better_strs) == 0:
+                print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset)
+                break
+
+            # Otherwise, save the new bitstring and repeat
+            best_indset, new_hamming_weight = better_strs[0]
+            best_init_state = cur_init_state
+            best_params = opt_params.copy()
+            best_perm = copy.copy(cur_permutation)
+            cur_init_state = best_indset
+            print('\tFound new independent set: {}, Hamming weight = {}'.format(
+                                               best_indset, new_hamming_weight))
+            
+            inner_round = inner_round+1
+        # Choose a new permutation of the mixer unitaries
+        cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
+            
+
+    print('\tRETURNING, best hamming weight:', new_hamming_weight)
+    return best_indset, best_params, best_init_state, best_perm
+
+
+######################################################################
+# Let's run this to find the MIS!
+# 
+
+base_str = '0'*len(graph.nodes)
+
+out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1)
+print('Init string: {}, Best MIS: {}'.format(''.join(base_str), out[0]))
+print()
+
+
+######################################################################
+# Starting with an all zero initial string will give us an independent
+# set, but this may not be the maximum. Let's try to run with with a few
+# different initial states. And also increase the cutoff value. ``cutoff``
+# indicates the number of bitstrings (sorted by the highest probabilities)
+# that we consider that may improve the Hamming weight.
+# 
+
+base_str = '0'*len(graph.nodes)
+for i in range(len(graph.nodes)):
+    init_str = list(base_str)
+    init_str[i] = '1'
+    out = solve_mis_qaoa(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=2)
+    print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0]))
+    print()
+
+
+######################################################################
+# DQVA
+# ----
+# 
+######################################################################
+# We will now formulate the MIS using the DQVA Ansatz. The cost function
+# is the same as the QAO-Ansatz (the Hamming weight operator).
+# 
+# In the DQVA, the way mixers are defined is slightly different from the
+# QAO-Ansatz and are allowed to be independent.
+# 
+# .. math::
+# 
+# 
+#    U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1)V_2^k(\alpha_k^2)\cdots V_N^k(\alpha_k^N))
+# 
+# where :math:`k = 1, 2 \cdots p`.
+# 
+# In the DQVA, whenever the :math:`j`-th bit of the initial state is one,
+# the corresponding parameter :math:`\alpha_k^j` is set to 0. For example,
+# if the initial state is :math:`\vert 01101 \rangle`, then
+# 
+# .. math::
+# 
+# 
+#    U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1) I_2 I_3 V_4^k(\alpha_k^4) I_5).
+# 
+# Therefore, we are dynamically turning off parameters thereby improving
+# the utilization of quantum resources.
+#
+
+def mixer_dqva(alpha, ancilla, init_state, mixer_order):
+    
+    # Permute the order of mixing unitaries
+    if mixer_order is None:
+        mixer_order = list(graph.nodes)
+    
     pad_alpha = [None]*len(init_state)
     next_alpha = 0
+    
     for qubit in mixer_order:
-        bit = list(reversed(init_state))[qubit]
+        bit = list(init_state)[qubit]
         if bit == '1' or next_alpha >= len(alpha):
             continue
         else:
             pad_alpha[qubit] = alpha[next_alpha]
             next_alpha += 1
-
-    print('init_state: {}, alpha: {}, pad_alpha: {}'.format(init_state,
-                                                              alpha, pad_alpha))
-
+        
     for qubit in mixer_order:
-        if pad_alpha[qubit] == None or not G.has_node(qubit):
+        
+        if pad_alpha[qubit] == None or not graph.has_node(qubit):
             # Turn off mixers for qubits which are already 1
             continue
-
-        neighbors = list(G.neighbors(qubit))
-        anc_idx = 4
-
-        qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors))
-
-        # apply an X rotation controlled by the state of the ancilla qubit
-        qml.CRX(2*pad_alpha[qubit], wires = [anc_idx, qubit])
+            
+        neighbors = list(graph.neighbors(qubit))
         
-        qml.MultiControlledX(control_wires= range(len(neighbors)),wires = len(neighbors), control_values ='0'*len(neighbors))
-
-def cost_layer(gamma, G):
-    """
-    Apply a parameterized Z-rotation to every qubit
-    """
-    for qb in G.nodes:
-        qml.RZ(2*gamma, wires=qb)
-
-def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None):
-    nq = len(G.nodes)
-    print(nq)
-
-    # Step 1: Jump Start
-    # Run an efficient classical approximation algorithm to warm-start the optimization
+        # Apply the multi-controlled Toffoli, targetting the ancilla qubit
+        ctrl_qubits = [i for i in neighbors]
+        qml.MultiControlledX(control_wires = ctrl_qubits,
+                             wires = [ancilla], 
+                             control_values ='0'*len(neighbors))
+        
+        qml.CRX(2*pad_alpha[qubit], wires=[ancilla, qubit])
+        
+        # Uncompute the ancilla
+        qml.MultiControlledX(control_wires = ctrl_qubits,
+                             wires = [ancilla], 
+                             control_values ='0'*len(neighbors))
+
+
+######################################################################
+# The structure of the cost and mixer unitaries is similar QAO-Ansatz,
+# however, we now alternate between :math:`p` applications of the mixing
+# unitary :math:`U_M^k(\alpha_k)` and :math:`p` applications of the cost
+# unitary :math:`U_C^k(\gamma_k)`:
+# 
+# .. math::
+# 
+# 
+#    \vert \alpha, \gamma \rangle = U_C^p(\gamma_p)U_M^p(\alpha_p)\cdots U_C^1(\gamma_p)U_M^1(\alpha_1)\vert c_1 \rangle
+# 
+# where :math:`\vert c_1 \rangle` is an initial state.
+# 
+
+def dqva_ansatz(P, params=[], init_state=None, mixer_order=None):
+    nq = len(graph.nodes)
+
+    # Step1: Initialize
     if init_state is None:
         init_state = '0'*nq
-
-    # Step 2: Mixer Initialization
-    for qb, bit in enumerate(reversed(init_state)):
-        if bit == '1':
-            qml.PauliX(wires = qb)
-            
-    # parse the variational parameters
-    # The dqva ansatz dynamically turns off partial mixers for qubits in |1>
-    # and adds extra mixers to the end of the circuit
-    num_nonzero = nq - hamming_weight(init_state)
-    assert (len(params) == (nq + 1) * P), "Incorrect number of parameters!"
+        
+    else:
+        for qb, bit in enumerate(reversed(init_state)):
+            if bit == '1':
+                qml.PauliX(wires=qb)
+    
+    num_nonzero = nq - hamming_weight(init_state)           
+    assert (len(params) == (nq + 1)*P),"Incorrect number of parameters!"
     alpha_list = []
     gamma_list = []
     last_idx = 0
@@ -144,30 +573,191 @@ def dqva_layer(G, P=1, params=[], init_state=None, mixer_order=None):
         alpha_list.append(cur_section[:-1])
         gamma_list.append(cur_section[-1])
         last_idx = (p+1)*chunk
-
-    # Add the leftover parameters as extra mixers
+    
     alpha_list.append(params[last_idx:])
-
-    for i in range(len(alpha_list)):
-        print('alpha_{}: {}'.format(i, alpha_list[i]))
-        if i < len(gamma_list):
-            print('gamma_{}: {}'.format(i, gamma_list[i]))
-
-    # Construct the dqva ansatz
-    #for alphas, gamma in zip(alpha_list, gamma_list):
+    
     for i in range(len(alpha_list)):
+        ancilla = nq
         alphas = alpha_list[i]
-        mixer_layer(alphas, init_state, G,  mixer_order)
-
+        mixer_dqva(alphas, ancilla, init_state, mixer_order)
 
         if i < len(gamma_list):
             gamma = gamma_list[i]
-            cost_layer(gamma, G)
-##############################################################################
-# Text in between code goes here
+            cost_layer(gamma)
+
+
+
+######################################################################
+# Let's also define the probability circuit for the DQVA ansatz
+# 
+
+@qml.qnode(dev)
+def probability_dqva(P, params=[], init_state=None, mixer_order=None):
+    dqva_ansatz(P, params, init_state, mixer_order)
+    return qml.probs(wires= dev.wires[:-1])
+
+
+######################################################################
+# The quantum-classical loop for DQVA also maximizes
+# 
+# .. math::
+# 
+# 
+#    \langle \alpha, \gamma \vert C_{obj} \vert \alpha, \gamma \rangle. 
+# 
+# The dynamic ansatz update includes: 1. Optimization of parameters using
+# Gradient Descent and finding the Hamming weight with new parameters 2.
+# If the Hamming weight of the new state is larger than the initial state
+# , the initial state gets updated to this new state 3. Based on this new
+# state, partial mixers are updated (i.e., turned off for ones and turned
+# on for zeros) 4. Steps 2 and 3 are repeated until the Hamming weight can
+# no longer be improved 5. If no new Hamming weight is obtained, the
+# partial mixers are randomized and steps 2 and 3 are repeated to check if
+# a better Hamming weight is found. The number of randomizations is
+# controlled via a hyperparameter.
+# 
+
+def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5,cutoff=1):
+    
+    # Select an ordering for the partial mixers
+    if mixer_order == None:
+        cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
+    else:
+        cur_permutation = mixer_order
 
+    def f(params):
+        
+        probs = probability_dqva(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation)
+        avg_cost = 0
+        for sample in range(0, len(probs)):
+            
 
+            x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))]
+            # Cost function is Hamming weight
+            avg_cost += probs[sample] * sum(x)
+        
+        return -avg_cost
+    
+    # Begin outer optimization loop
+    best_indset = init_state
+    best_init_state = init_state
+    cur_init_state = init_state
+    best_params = None
+    best_perm = copy.copy(cur_permutation)
+    
+    # Randomly permute the order of mixer unitaries m times
+    for mixer_round in range (1, m+1):
 
+        inner_round = 1
+        new_hamming_weight = hamming_weight(cur_init_state)
+        
+        while True:
+            print('Start round {}.{}, Initial state = {}'.format(mixer_round,
+                inner_round, cur_init_state))
+            
+            # Begin inner variational loop
+            num_params = P * (len(graph.nodes()) + 1)
+            print('\tNum params =', num_params)
+            init_params = np.random.uniform(low=0.0, high=2*np.pi, size=num_params)
+            print('\tCurrent Mixer Order:', cur_permutation)
+            
+            # Optimize parameters
+            optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
+            cur_params = init_params.copy()
+            
+            for i in range(70):
+                cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
+
+            opt_params = cur_params.copy()
+            
+            print('\tOptimal cost:', opt_cost)
+            
+            probs = probability_dqva(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation)
+
+                    
+            top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1]
+
+            best_hamming_weight = hamming_weight(best_indset)
+            better_strs = []
+            
+            for bitstr in top_counts[:1]:
+                this_hamming = hamming_weight(bitstr)
+                if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
+                    better_strs.append((bitstr, this_hamming))
+            better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
+        
+
+            # If no improvement was made, break and go to next mixer round
+            if len(better_strs) == 0:
+                print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset)
+                break
+
+            # Otherwise, save the new bitstring and repeat
+            best_indset, new_hamming_weight = better_strs[0]
+            best_init_state = cur_init_state
+            best_params = opt_params
+            best_perm = copy.copy(cur_permutation)
+            cur_init_state = best_indset
+            print('\tFound new independent set: {}, Hamming weight = {}'.format(
+                                               best_indset, new_hamming_weight))
+            inner_round += 1
+
+
+        # Choose a new permutation of the mixer unitaries
+        cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
+
+    print('\tRETURNING, best hamming weight:', new_hamming_weight)
+    return best_indset, best_params, best_init_state, best_perm
+
+
+######################################################################
+# We also run the algorithm for a number of initial states (instead of
+# just one) to check what is the best Maximum Independent Set we obtain.
+# The initial states are a set of independent sets of size one (this is
+# trivial - any bitstring with a single "1" is a valid independent set)
+# 
+
+base_str = '0'*len(graph.nodes)
+all_init_strs = []
+for i in range(len(graph.nodes)):
+    init_str = list(base_str)
+    init_str[i] = '1'
+    out = solve_mis_dqva(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=1)
+    print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0]))
+    print()
+
+
+######################################################################
+# And voila, we have found an independent set of size 3 which is the state
+# ``10101``!
+# 
+
+
+######################################################################
+# Conclusion
+# ----------
+# 
+######################################################################
+# In this demo, we looked into solving a constrained combinatorial
+# optimization problem using the QAO-Ansatz and the DQVA. The QAO-Ansatz
+# defines mixer unitaries with partial mixers in such a way that we never
+# leave the set of feasible states. However, this comes at a high cost of
+# using multi-controlled Toffoli gates. To overcome this, we looked at the
+# DQVA which turned partial mixers on and off, and randomized the partial
+# mixer ordering. Furthermore, Saleem et al., recommend starting with a
+# set of initial states that are independent sets found by any classical
+# polynomial time approximation algorithm to "warm start" the
+# initialization. Furthermore, they propose the first useful application
+# of `circuit-cutting techniques <https://arxiv.org/abs/2107.07532>`__ to
+# solve MIS, which involves classical partitioning of a large graph into
+# sub-graphs, finding a solution using DQVA on a subgraph, and then
+# preparing a set of states to be passed as an input, essentially
+# stitching together solutions.
+# 
+
+
+
+######################################################################
 #
 # References
 # ----------
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
new file mode 100644
index 0000000000..113d2a2b94
--- /dev/null
+++ b/test-results/sphinx-gallery/junit.xml
@@ -0,0 +1 @@
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="0.15500259399414062"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Solving MIS with the DQVA Ansatz" time="0.15500259399414062"></testcase></testsuite>
\ No newline at end of file

From c2ecb63fb8fa701555603c5e8d4140debbe34ad9 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Fri, 14 Jan 2022 23:22:56 -0800
Subject: [PATCH 04/28] minor edits

---
 demonstrations/tutorial_dqva_mis.py | 513 +++++++++++++++-------------
 1 file changed, 272 insertions(+), 241 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 9dba349f1b..d6e4a13e9b 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -23,23 +23,23 @@
 ######################################################################
 # QAOA, QAO-Ansatz, and DQVA
 # --------------------------
-# 
+#
 # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid
 # quantum-classical algorithm by `Farhi et
 # al. <https://arxiv.org/abs/1411.4028>`__ is a significant meta-heuristic
 # for finding solutions to combinatorial optimization problems. The
 # quantum part involves alternating between a *cost Hamiltonian*,
-# :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evalulate the
+# :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evaluate the
 # objective function and a classical optimization loop updates the ansatz
 # parameters. You can learn more about this in `PennyLane's tutorial on
 # QAOA <https://pennylane.ai/qml/demos/tutorial_qaoa_intro.html>`__.
-# 
+#
 # For solving constrained combinatorial optimization problems such as MIS
-# using QAOA, constraints can be imposed in two ways: 1. By adding a
-# penalty term to the problem's objective function (the Lagrange
-# multiplier approach) 2. By constructing the variational ansatz in a way
-# such that constraints are satisfied at all times
-# 
+# using QAOA, constraints can be imposed in two ways:
+#
+# 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach)
+# 2. By constructing the variational ansatz in a way such that constraints are satisfied at all times
+#
 # Hadfield et al., extend the QAOA framework and introduce the **Quantum
 # Alternating Operator Ansatz (QAO-Ansatz)** which alternates between more
 # general families of operators thereby supporting potentially more useful
@@ -47,15 +47,15 @@
 # example, a mixing operator :math:`U_M (\beta)` is decomposed into a
 # sequence of partial mixers :math:`U_{M, x}` which may not correspond to
 # time evolution under a fixed mixer Hamiltonian :math:`H_M`.
-# 
-# .. figure:: ../demonstrations/dqva_mis/independent_sets.png
+#
+# .. figure:: ../demonstrations/dqva_mis/partial_mixers.png
 #    :align: center
 #    :width: 100%
 #
 #    ..
 #
 #    Independent sets of a graph
-# 
+#
 # The QAO-Ansatz guarantees that the output never leaves the set of
 # feasible states (in contrast, the penalty term optimization requires an
 # additional pruning step since the output contains both feasible and
@@ -64,34 +64,34 @@
 # Multi-Controlled Toffoli gates that require high connectivity between
 # qubits. This makes the QAO-Ansatz inadequate for large graphs on current
 # near-term quantum computers.
-# 
+#
 # This brings us to the **Dynamic Quantum Variational Ansatz (DQVA)** that
-# maximizes the performance of the QAO-Ansatz by: 
-# 
+# maximizes the performance of the QAO-Ansatz by:
+#
 # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state
 # or a superposition of feasible states. A feasible state can be found by
-# using a classical approximate polynomial time algorithm. 
-# 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off* 
+# using a classical approximate polynomial-time algorithm.
+# 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off*
 # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries
-# 
+#
 # Now that we have the basics in place, let's implement the QAO-Ansatz and
 # the DQVA!
 
-# 
+#
 
 
 ######################################################################
 # Formulating the Maximum Independent Set (MIS) problem
 # -----------------------------------------------------
-# 
+#
 # Given a graph, :math:`G = (V, E)` where :math:`V` is the set of vertices
 # and :math:`E` is the set of edges, an *independent set* is a subset of
 # vertices, :math:`V' \subset V`, such that no two vertices in :math:`V'`
-# share an edge. A *Maximum Independet Set* (MIS) is an independent set of
+# share an edge. A *Maximum Independent Set* (MIS) is an independent set of
 # the largest possible size. In this demo, our objective is to find the
 # MIS of a given graph. The following image shows some independent sets of
 # a graph with 5 vertices:
-# 
+#
 # .. figure:: ../demonstrations/dqva_mis/independent_sets.png
 #    :align: center
 #    :width: 100%
@@ -99,18 +99,18 @@
 #    ..
 #
 #    Independent sets of a graph
-# 
+#
 # Here, independent sets can be represented as bitstrings
 # :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown
 # above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are
 # independent sets of size 0, 1, and 2 respectively. The Minimum Vertex
 # Cover, i.e, vertices that cover all edges, is the dual of MIS.
-# 
+#
 # We will start with the QAO-Ansatz and then re-use most components to
 # formulate the DQVA.
-# 
+#
 # First, we start with the necessary imports.
-# 
+#
 
 import pennylane as qml
 from pennylane import qaoa
@@ -123,7 +123,7 @@
 ######################################################################
 # Next, we define the graph shown above with 5 nodes for which we would
 # like to find the Maximum Independent Set.
-# 
+#
 
 edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)]
 # edges = [(0, 1), (0, 3), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)]
@@ -135,7 +135,7 @@
 ######################################################################
 # We will also instantiate a device with the number of wires being one
 # more than the number of nodes (the extra wire is an ancilla).
-# 
+#
 
 wires = len(graph.nodes) + 1
 dev = qml.device("default.qubit", wires=range(wires))
@@ -147,38 +147,40 @@
 # weight <https://en.wikipedia.org/wiki/Hamming_weight>`__ of the
 # bitstring. Therefore, our objective function is the Hamming weight
 # operator,
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    C_{obj} = H = \sum_{i \in V} b_i,
-# 
+#
 # where
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    b_i = \frac{1}{2} (1 - Z_i).
-# 
+#
 # Here :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th
 # qubit.
-# 
+#
 
 
 ######################################################################
 # We need a couple of helper functions here: ``hamming_weight``, which
 # calculates the Hamming weight of a given bitstring and ``is_indset``,
 # which verifies whether a given bitstring is a valid independent set.
-# 
+#
+
 
 def hamming_weight(bitstr):
-    return sum([1 for bit in bitstr if bit == '1'])
+    return sum([1 for bit in bitstr if bit == "1"])
+
 
 def is_indset(bitstr, G):
     nodes = list(G.nodes)
     ind_set = []
     for idx, bit in enumerate(bitstr):
-        if bit == '1':
+        if bit == "1":
             cur_neighbors = list(G.neighbors(idx))
             for node in ind_set:
                 if node in cur_neighbors:
@@ -191,74 +193,75 @@ def is_indset(bitstr, G):
 ######################################################################
 # Next, we define a cost unitary that incorporates the objective function,
 # :math:`H` and is parameterized by :math:`\gamma`:
-# 
+#
 # .. math::
-# 
-# 
-#    U_C(\gamma) = \prod_i e^{i \gamma H}. 
-# 
+#
+#
+#    U_C(\gamma) = \prod_i e^{i \gamma H}.
+#
+
 
 def cost_layer(gamma):
     for qb in graph.nodes:
-        qml.RZ(2*gamma, wires=qb)
+        qml.RZ(2 * gamma, wires=qb)
 
 
 ######################################################################
 # The mixer unitary for the QAO-Ansatz is defined as $
 # U_M(:raw-latex:`\beta`) = :raw-latex:`\prod`\_i
 # e^{i:raw-latex:`\beta `M_i}$
-# 
+#
 # where
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    M_i = X_i \tilde{B},
-# 
+#
 # and
-# 
+#
 # .. math::
 #
-# 
+#
 #    \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}.
-# 
+#
 # Here, where :math:`v_j` are the neighbors and :math:`l` is the number of
 # neighbors for the :math:`i`-th node and
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    \tilde{b}_{v_j} = \frac{1+Z_{v_j}}{2}.
-# 
+#
 # The mixer unitary can also be written as a product of :math:`N` partial
 # mixers :math:`V_i`:
-# 
+#
 # .. math::
-# 
-# 
-#    U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N  (I + (e^{-i\beta X_i} - I)\tilde{B}) $
-# 
+#
+#
+#    U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N  (I + (e^{-i\beta X_i} - I)\tilde{B}) 
+#
 # The partial mixers may not commute with each other, i.e.,
 # :math:`[V_i, V_j] \neq 0` therefore, different orderings of these mixers
 # may have different outputs. Therefore, the variational ansatz is defined
 # up to a permutation:
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_N(\beta))
-# 
+#
 # where P is the permutation's function of labels from :math:`1` to
 # :math:`N`.
-# 
+#
 # These partial mixers are implemented using parameterized
 # controlled-:math:`X` rotations sandwiched between two `multi-controlled
 # Toffoli
 # gates <https://pennylane.readthedocs.io/en/stable/code/api/pennylane.MultiControlledX.html>`__.
 # The number of qubits we require is one more than the number of vertices.
-# This extra qubit is used as an ancilliary qubit to serve as the target
+# This extra qubit is used as an ancillary qubit to serve as the target
 # for the multi-controlled Toffoli gates.
-# 
+#
 # .. figure:: ../demonstrations/dqva_mis/mixer_unitary.png
 #    :align: center
 #    :width: 100%
@@ -266,68 +269,71 @@ def cost_layer(gamma):
 #    ..
 #
 #    Formulation of the mixer unitary
-# 
+#
+# In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates
+# controlled by the neighbors of the :math:`i`'th vertex.
 
 def mixer_layer(beta, ancilla, mixer_order):
-    
+
     # Permute the order of mixing unitaries
     if mixer_order is None:
         mixer_order = list(graph.nodes)
-    
+
     for qubit in list(mixer_order):
         neighbors = list(graph.neighbors(qubit))
 
         # Apply the multi-controlled Toffoli, targetting the ancilla qubit
         ctrl_qubits = [i for i in neighbors]
-        qml.MultiControlledX(control_wires = ctrl_qubits,
-                             wires = [ancilla], 
-                             control_values ='0'*len(neighbors))
-        
-        qml.CRX(2*beta, wires=[ancilla, qubit])
-        
+        qml.MultiControlledX(
+            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
+        )
+
+        qml.CRX(2 * beta, wires=[ancilla, qubit])
+
         # Uncompute the ancilla
-        qml.MultiControlledX(control_wires = ctrl_qubits,
-                             wires = [ancilla], 
-                             control_values ='0'*len(neighbors))
+        qml.MultiControlledX(
+            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
+        )
 
 
 ######################################################################
 # We are now ready to build the circuit consisting of alternating cost and
 # mixer layers. A list of ``params`` includes parameters for the mixer
 # layer (even elements) and cost layer (odd elements).
-# 
+#
+
 
 def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
     nq = len(graph.nodes)
     # Initialize
     if init_state is None:
-        init_state = '0'*nq
-        
+        init_state = "0" * nq
+
     else:
         for qb, bit in enumerate(reversed(init_state)):
-            if bit == '1':
+            if bit == "1":
                 qml.PauliX(wires=qb)
-    assert (len(params) == 2*P),"Incorrect number of parameters!"
-    
-    betas  = [a for i, a in enumerate(params) if i % 2 == 0]
+    assert len(params) == 2 * P, "Incorrect number of parameters!"
+
+    betas = [a for i, a in enumerate(params) if i % 2 == 0]
     gammas = [a for i, a in enumerate(params) if i % 2 == 1]
-    
+
     for beta, gamma in zip(betas, gammas):
-        ancilla = nq # Use the last qubit as an ancilla
+        ancilla = nq  # Use the last qubit as an ancilla
         mixer_layer(beta, ancilla, mixer_order)
         cost_layer(gamma)
 
 
-
 ######################################################################
 # Now, let's build the circuit that finds the probabilities of measuring
 # each bitstring.
-# 
+#
+
 
 @qml.qnode(dev)
 def probability_circuit(P, params=[], init_state=None, mixer_order=None):
     qaoa_ansatz(P, params, init_state, mixer_order)
-    return qml.probs(wires=range(wires-1))
+    return qml.probs(wires=range(wires - 1))
 
 
 ######################################################################
@@ -339,12 +345,13 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None):
 # from classical minimization of the cost function (Gradient Descent)
 #
 # .. math::
-# 
-# 
-#    \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle. 
-# 
+#
+#
+#    \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle.
+#
+
 
-def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None,  threshold=1e-5, cutoff=1):
+def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1):
     # Select an ordering for the partial mixers
     if mixer_order == None:
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
@@ -354,70 +361,82 @@ def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None,  threshold=1e-5, cuto
     # Define the function to be optimized
     # Note that we are returning -cost since this is a minimization
     def f(params):
-        probs = probability_circuit(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation)
+        probs = probability_circuit(
+            P, params=params, init_state=cur_init_state, mixer_order=cur_permutation
+        )
         avg_cost = 0
         for sample in range(0, len(probs)):
-            
+
             x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))]
             # Cost function is Hamming weight
             avg_cost += probs[sample] * sum(x)
-        
+
         return -avg_cost
-    
+
     # Begin outer optimization loop
     best_indset = init_state
     best_init_state = init_state
     cur_init_state = init_state
     best_params = None
     best_perm = copy.copy(cur_permutation)
-    
+
     # Randomly permute the order of mixer unitaries m times
-    for mixer_round in range (1, m+1):
-     
+    for mixer_round in range(1, m + 1):
+
         new_hamming_weight = hamming_weight(cur_init_state)
         inner_round = 1
-         
-        while (inner_round < 2):
-            print('Start round {}.{}, Initial state = {}'.format(mixer_round, inner_round, cur_init_state))
+
+        while inner_round < 2:
+            print(
+                "Start round {}.{}, Initial state = {}".format(
+                    mixer_round, inner_round, cur_init_state
+                )
+            )
 
             # Begin inner variational loop
             num_params = 2 * P
-            print('\tNum params =', num_params)
+            print("\tNum params =", num_params)
 
             init_params = np.random.uniform(low=-np.pi, high=np.pi, size=num_params)
-            print('\tCurrent Mixer Order:', cur_permutation)
+            print("\tCurrent Mixer Order:", cur_permutation)
 
             # Optimize parameters
             optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
             cur_params = init_params.copy()
-            
+
             for i in range(70):
                 cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
 
             opt_params = cur_params.copy()
 
-            print('\tOptimal cost:', opt_cost)
+            print("\tOptimal cost:", opt_cost)
 
             # Obtain probabilites
-            probs = probability_circuit(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation)
-            top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1]
+            probs = probability_circuit(
+                P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation
+            )
+            top_counts = list(
+                map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
+            )[::-1]
             # print (top_counts)
-            
+
             best_hamming_weight = hamming_weight(best_indset)
             better_strs = []
-            
+
             print(top_counts[:cutoff])
             for bitstr in top_counts[:cutoff]:
-                
+
                 this_hamming = hamming_weight(bitstr)
                 if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
                     better_strs.append((bitstr, this_hamming))
-                    
+
             better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
 
             # If no improvement was made, break and go to next mixer round
             if len(better_strs) == 0:
-                print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset)
+                print(
+                    "\tNone of the measured bitstrings had higher Hamming weight than:", best_indset
+                )
                 break
 
             # Otherwise, save the new bitstring and repeat
@@ -426,114 +445,117 @@ def f(params):
             best_params = opt_params.copy()
             best_perm = copy.copy(cur_permutation)
             cur_init_state = best_indset
-            print('\tFound new independent set: {}, Hamming weight = {}'.format(
-                                               best_indset, new_hamming_weight))
-            
-            inner_round = inner_round+1
+            print(
+                "\tFound new independent set: {}, Hamming weight = {}".format(
+                    best_indset, new_hamming_weight
+                )
+            )
+
+            inner_round = inner_round + 1
         # Choose a new permutation of the mixer unitaries
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
-            
 
-    print('\tRETURNING, best hamming weight:', new_hamming_weight)
+    print("\tRETURNING, best hamming weight:", new_hamming_weight)
     return best_indset, best_params, best_init_state, best_perm
 
 
 ######################################################################
 # Let's run this to find the MIS!
-# 
+#
 
-base_str = '0'*len(graph.nodes)
+base_str = "0" * len(graph.nodes)
 
 out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1)
-print('Init string: {}, Best MIS: {}'.format(''.join(base_str), out[0]))
+print("Init string: {}, Best MIS: {}".format("".join(base_str), out[0]))
 print()
 
 
 ######################################################################
-# Starting with an all zero initial string will give us an independent
-# set, but this may not be the maximum. Let's try to run with with a few
-# different initial states. And also increase the cutoff value. ``cutoff``
+# Starting with an all-zero initial string will give us an independent
+# set, but this may not be the maximum. Now, we run this with a few
+# different initial states and also increase the cutoff value. ``cutoff``
 # indicates the number of bitstrings (sorted by the highest probabilities)
 # that we consider that may improve the Hamming weight.
-# 
+#
 
-base_str = '0'*len(graph.nodes)
+base_str = "0" * len(graph.nodes)
 for i in range(len(graph.nodes)):
     init_str = list(base_str)
-    init_str[i] = '1'
-    out = solve_mis_qaoa(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=2)
-    print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0]))
+    init_str[i] = "1"
+    out = solve_mis_qaoa("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=2)
+    print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0]))
     print()
 
 
 ######################################################################
 # DQVA
 # ----
-# 
+#
 ######################################################################
 # We will now formulate the MIS using the DQVA Ansatz. The cost function
 # is the same as the QAO-Ansatz (the Hamming weight operator).
-# 
+#
 # In the DQVA, the way mixers are defined is slightly different from the
 # QAO-Ansatz and are allowed to be independent.
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1)V_2^k(\alpha_k^2)\cdots V_N^k(\alpha_k^N))
-# 
+#
 # where :math:`k = 1, 2 \cdots p`.
-# 
+#
 # In the DQVA, whenever the :math:`j`-th bit of the initial state is one,
 # the corresponding parameter :math:`\alpha_k^j` is set to 0. For example,
 # if the initial state is :math:`\vert 01101 \rangle`, then
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1) I_2 I_3 V_4^k(\alpha_k^4) I_5).
-# 
+#
 # Therefore, we are dynamically turning off parameters thereby improving
 # the utilization of quantum resources.
 #
 
+
 def mixer_dqva(alpha, ancilla, init_state, mixer_order):
-    
+
     # Permute the order of mixing unitaries
     if mixer_order is None:
         mixer_order = list(graph.nodes)
-    
-    pad_alpha = [None]*len(init_state)
+
+    pad_alpha = [None] * len(init_state)
     next_alpha = 0
-    
+
     for qubit in mixer_order:
         bit = list(init_state)[qubit]
-        if bit == '1' or next_alpha >= len(alpha):
+        if bit == "1" or next_alpha >= len(alpha):
             continue
         else:
             pad_alpha[qubit] = alpha[next_alpha]
             next_alpha += 1
-        
+
     for qubit in mixer_order:
-        
+
         if pad_alpha[qubit] == None or not graph.has_node(qubit):
             # Turn off mixers for qubits which are already 1
             continue
-            
+
         neighbors = list(graph.neighbors(qubit))
-        
+
         # Apply the multi-controlled Toffoli, targetting the ancilla qubit
         ctrl_qubits = [i for i in neighbors]
-        qml.MultiControlledX(control_wires = ctrl_qubits,
-                             wires = [ancilla], 
-                             control_values ='0'*len(neighbors))
-        
-        qml.CRX(2*pad_alpha[qubit], wires=[ancilla, qubit])
-        
+        qml.MultiControlledX(
+            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
+        )
+
+        qml.CRX(2 * pad_alpha[qubit], wires=[ancilla, qubit])
+
         # Uncompute the ancilla
-        qml.MultiControlledX(control_wires = ctrl_qubits,
-                             wires = [ancilla], 
-                             control_values ='0'*len(neighbors))
+        qml.MultiControlledX(
+            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
+        )
 
 
 ######################################################################
@@ -541,41 +563,42 @@ def mixer_dqva(alpha, ancilla, init_state, mixer_order):
 # however, we now alternate between :math:`p` applications of the mixing
 # unitary :math:`U_M^k(\alpha_k)` and :math:`p` applications of the cost
 # unitary :math:`U_C^k(\gamma_k)`:
-# 
+#
 # .. math::
-# 
-# 
+#
+#
 #    \vert \alpha, \gamma \rangle = U_C^p(\gamma_p)U_M^p(\alpha_p)\cdots U_C^1(\gamma_p)U_M^1(\alpha_1)\vert c_1 \rangle
-# 
+#
 # where :math:`\vert c_1 \rangle` is an initial state.
-# 
+#
+
 
 def dqva_ansatz(P, params=[], init_state=None, mixer_order=None):
     nq = len(graph.nodes)
 
     # Step1: Initialize
     if init_state is None:
-        init_state = '0'*nq
-        
+        init_state = "0" * nq
+
     else:
         for qb, bit in enumerate(reversed(init_state)):
-            if bit == '1':
+            if bit == "1":
                 qml.PauliX(wires=qb)
-    
-    num_nonzero = nq - hamming_weight(init_state)           
-    assert (len(params) == (nq + 1)*P),"Incorrect number of parameters!"
+
+    num_nonzero = nq - hamming_weight(init_state)
+    assert len(params) == (nq + 1) * P, "Incorrect number of parameters!"
     alpha_list = []
     gamma_list = []
     last_idx = 0
     for p in range(P):
         chunk = num_nonzero + 1
-        cur_section = params[p*chunk:(p+1)*chunk]
+        cur_section = params[p * chunk : (p + 1) * chunk]
         alpha_list.append(cur_section[:-1])
         gamma_list.append(cur_section[-1])
-        last_idx = (p+1)*chunk
-    
+        last_idx = (p + 1) * chunk
+
     alpha_list.append(params[last_idx:])
-    
+
     for i in range(len(alpha_list)):
         ancilla = nq
         alphas = alpha_list[i]
@@ -586,39 +609,38 @@ def dqva_ansatz(P, params=[], init_state=None, mixer_order=None):
             cost_layer(gamma)
 
 
-
 ######################################################################
 # Let's also define the probability circuit for the DQVA ansatz
-# 
+#
+
 
 @qml.qnode(dev)
 def probability_dqva(P, params=[], init_state=None, mixer_order=None):
     dqva_ansatz(P, params, init_state, mixer_order)
-    return qml.probs(wires= dev.wires[:-1])
+    return qml.probs(wires=dev.wires[:-1])
 
 
 ######################################################################
 # The quantum-classical loop for DQVA also maximizes
-# 
+#
 # .. math::
-# 
-# 
-#    \langle \alpha, \gamma \vert C_{obj} \vert \alpha, \gamma \rangle. 
-# 
-# The dynamic ansatz update includes: 1. Optimization of parameters using
-# Gradient Descent and finding the Hamming weight with new parameters 2.
-# If the Hamming weight of the new state is larger than the initial state
-# , the initial state gets updated to this new state 3. Based on this new
-# state, partial mixers are updated (i.e., turned off for ones and turned
-# on for zeros) 4. Steps 2 and 3 are repeated until the Hamming weight can
-# no longer be improved 5. If no new Hamming weight is obtained, the
-# partial mixers are randomized and steps 2 and 3 are repeated to check if
-# a better Hamming weight is found. The number of randomizations is
-# controlled via a hyperparameter.
-# 
-
-def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5,cutoff=1):
-    
+#
+#
+#    \langle \alpha, \gamma \vert C_{obj} \vert \alpha, \gamma \rangle.
+#
+# The dynamic ansatz update includes:
+#
+# 1. Optimization of parameters using Gradient Descent and finding the Hamming weight with new parameters
+# 2. If the Hamming weight of the new state is larger than the initial state, the initial state gets updated to this new state
+# 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros)
+# 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved
+# 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if
+# a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
+#
+
+
+def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1):
+
     # Select an ordering for the partial mixers
     if mixer_order == None:
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
@@ -626,70 +648,78 @@ def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5,cutoff
         cur_permutation = mixer_order
 
     def f(params):
-        
-        probs = probability_dqva(P, params=params, init_state=cur_init_state, mixer_order=cur_permutation)
+
+        probs = probability_dqva(
+            P, params=params, init_state=cur_init_state, mixer_order=cur_permutation
+        )
         avg_cost = 0
         for sample in range(0, len(probs)):
-            
 
             x = [int(bit) for bit in list(np.binary_repr(sample, len(graph.nodes)))]
             # Cost function is Hamming weight
             avg_cost += probs[sample] * sum(x)
-        
+
         return -avg_cost
-    
+
     # Begin outer optimization loop
     best_indset = init_state
     best_init_state = init_state
     cur_init_state = init_state
     best_params = None
     best_perm = copy.copy(cur_permutation)
-    
+
     # Randomly permute the order of mixer unitaries m times
-    for mixer_round in range (1, m+1):
+    for mixer_round in range(1, m + 1):
 
         inner_round = 1
         new_hamming_weight = hamming_weight(cur_init_state)
-        
+
         while True:
-            print('Start round {}.{}, Initial state = {}'.format(mixer_round,
-                inner_round, cur_init_state))
-            
+            print(
+                "Start round {}.{}, Initial state = {}".format(
+                    mixer_round, inner_round, cur_init_state
+                )
+            )
+
             # Begin inner variational loop
             num_params = P * (len(graph.nodes()) + 1)
-            print('\tNum params =', num_params)
-            init_params = np.random.uniform(low=0.0, high=2*np.pi, size=num_params)
-            print('\tCurrent Mixer Order:', cur_permutation)
-            
+            print("\tNum params =", num_params)
+            init_params = np.random.uniform(low=0.0, high=2 * np.pi, size=num_params)
+            print("\tCurrent Mixer Order:", cur_permutation)
+
             # Optimize parameters
             optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
             cur_params = init_params.copy()
-            
+
             for i in range(70):
                 cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
 
             opt_params = cur_params.copy()
-            
-            print('\tOptimal cost:', opt_cost)
-            
-            probs = probability_dqva(P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation)
 
-                    
-            top_counts = list(map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs)))[::-1]
+            print("\tOptimal cost:", opt_cost)
+
+            probs = probability_dqva(
+                P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation
+            )
+
+            top_counts = list(
+                map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
+            )[::-1]
 
             best_hamming_weight = hamming_weight(best_indset)
             better_strs = []
-            
+
             for bitstr in top_counts[:1]:
                 this_hamming = hamming_weight(bitstr)
                 if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
                     better_strs.append((bitstr, this_hamming))
             better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
-        
 
             # If no improvement was made, break and go to next mixer round
             if len(better_strs) == 0:
-                print('\tNone of the measured bitstrings had higher Hamming weight than:', best_indset)
+                print(
+                    "\tNone of the measured bitstrings had higher Hamming weight than:", best_indset
+                )
                 break
 
             # Otherwise, save the new bitstring and repeat
@@ -698,45 +728,47 @@ def f(params):
             best_params = opt_params
             best_perm = copy.copy(cur_permutation)
             cur_init_state = best_indset
-            print('\tFound new independent set: {}, Hamming weight = {}'.format(
-                                               best_indset, new_hamming_weight))
+            print(
+                "\tFound new independent set: {}, Hamming weight = {}".format(
+                    best_indset, new_hamming_weight
+                )
+            )
             inner_round += 1
 
-
         # Choose a new permutation of the mixer unitaries
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
 
-    print('\tRETURNING, best hamming weight:', new_hamming_weight)
+    print("\tRETURNING, best hamming weight:", new_hamming_weight)
     return best_indset, best_params, best_init_state, best_perm
 
 
 ######################################################################
-# We also run the algorithm for a number of initial states (instead of
+# We also run the algorithm for several initial states (instead of
 # just one) to check what is the best Maximum Independent Set we obtain.
 # The initial states are a set of independent sets of size one (this is
 # trivial - any bitstring with a single "1" is a valid independent set)
-# 
+#
 
-base_str = '0'*len(graph.nodes)
+base_str = "0" * len(graph.nodes)
 all_init_strs = []
 for i in range(len(graph.nodes)):
     init_str = list(base_str)
-    init_str[i] = '1'
-    out = solve_mis_dqva(''.join(init_str), P=1, m=4, threshold=1e-5, cutoff=1)
-    print('Init string: {}, Best MIS: {}'.format(''.join(init_str), out[0]))
+    init_str[i] = "1"
+    out = solve_mis_dqva("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=1)
+    print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0]))
     print()
 
 
 ######################################################################
 # And voila, we have found an independent set of size 3 which is the state
 # ``10101``!
-# 
+#
 
 
 ######################################################################
 # Conclusion
 # ----------
-# 
+#
 ######################################################################
 # In this demo, we looked into solving a constrained combinatorial
 # optimization problem using the QAO-Ansatz and the DQVA. The QAO-Ansatz
@@ -753,8 +785,7 @@ def f(params):
 # sub-graphs, finding a solution using DQVA on a subgraph, and then
 # preparing a set of states to be passed as an input, essentially
 # stitching together solutions.
-# 
-
+#
 
 
 ######################################################################

From 7220453aa61672fbc954d588f0df721e96ff4b6f Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Sat, 15 Jan 2022 11:27:37 -0800
Subject: [PATCH 05/28] fix errors

---
 demonstrations/tutorial_dqva_mis.py   | 12 +++---------
 test-results/sphinx-gallery/junit.xml |  2 +-
 2 files changed, 4 insertions(+), 10 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index d6e4a13e9b..4721da2a30 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -4,7 +4,7 @@
 ============================================
 
 .. meta::
-    :property="og:description": Description and assessment of trapped ion quantum computers
+    :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA
     :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png
 
 .. related::
@@ -14,9 +14,6 @@
 *Author: Priya Angara*
 
 This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) <https://arxiv.org/abs/2010.06660>` and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) <https://arxiv.org/abs/1709.03489)>` for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. 
-.. container:: alert alert-block alert-info
-    
-    Alert block
 
 """
 
@@ -488,10 +485,8 @@ def f(params):
 
 
 ######################################################################
-# DQVA
-# ----
-#
-######################################################################
+# Dynamic Quantum Variational Ansatz
+# ----------------------------------
 # We will now formulate the MIS using the DQVA Ansatz. The cost function
 # is the same as the QAO-Ansatz (the Hamming weight operator).
 #
@@ -769,7 +764,6 @@ def f(params):
 # Conclusion
 # ----------
 #
-######################################################################
 # In this demo, we looked into solving a constrained combinatorial
 # optimization problem using the QAO-Ansatz and the DQVA. The QAO-Ansatz
 # defines mixer unitaries with partial mixers in such a way that we never
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 113d2a2b94..7fd1edd634 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1 +1 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="0.15500259399414062"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Solving MIS with the DQVA Ansatz" time="0.15500259399414062"></testcase></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="0" time="0.0"></testsuite>
\ No newline at end of file

From cafd6825092f29083a16282887347489fc9b5897 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Sat, 15 Jan 2022 18:01:57 -0800
Subject: [PATCH 06/28] update toc

---
 demonstrations/tutorial_dqva_mis.py | 2 +-
 demos_optimization.rst              | 7 +++++++
 2 files changed, 8 insertions(+), 1 deletion(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 4721da2a30..aee60c882c 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -5,7 +5,7 @@
 
 .. meta::
     :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA
-    :property="og:image": https://pennylane.ai/qml/_images/trapped_ions_tn.png
+    :property="og:image": https://pennylane.ai/qml/_images/mixer_unitary.png
 
 .. related::
    tutorial_qaoa_intro Introduction to QAOA
diff --git a/demos_optimization.rst b/demos_optimization.rst
index 26704466c0..46908048de 100644
--- a/demos_optimization.rst
+++ b/demos_optimization.rst
@@ -139,6 +139,12 @@ in quantum neural networks.
     :description: :doc:`demos/tutorial_qaoa_maxcut`
     :tags: autograd
 
+.. customgalleryitem::
+    :tooltip: QAO-Ansatz and DQVA for MIS.
+    :figure: demonstrations/dqva_mis/mixer-unitary.png
+    :description: :doc:`demos/tutorial_dqva_mis`
+    :tags: autograd
+
 :html:`</div></div><div style='clear:both'>`
 
 
@@ -166,3 +172,4 @@ in quantum neural networks.
     demos/qonn
     demos/tutorial_isingmodel_PyTorch
     demos/tutorial_qaoa_maxcut
+    demos/tutorial_dqva_mis
\ No newline at end of file

From 721673f1565594c1fdfa540b23824ec9190b35e8 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Sun, 16 Jan 2022 09:31:45 -0800
Subject: [PATCH 07/28] fix latex issues

---
 demonstrations/tutorial_dqva_mis.py | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index aee60c882c..b653582592 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -204,9 +204,12 @@ def cost_layer(gamma):
 
 
 ######################################################################
-# The mixer unitary for the QAO-Ansatz is defined as $
-# U_M(:raw-latex:`\beta`) = :raw-latex:`\prod`\_i
-# e^{i:raw-latex:`\beta `M_i}$
+# The mixer unitary for the QAO-Ansatz is defined as 
+#
+# .. math::
+#
+#
+#       U_M(\beta) = \prod\_i e^{i \beta M_i}
 #
 # where
 #
@@ -787,8 +790,6 @@ def f(params):
 # References
 # ----------
 #
-# .. [#Saleem2020]
-#
-#     Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization",
+# .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization",
 #     `arXiv preprint arXiv:2010.06660 <https://arxiv.org/abs/2010.06660>`__.
 #

From 428cf66bb9906aca8c05b6896dd9cf01020f1111 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Sun, 16 Jan 2022 10:17:27 -0800
Subject: [PATCH 08/28] fix figure

---
 demonstrations/tutorial_dqva_mis.py | 16 +++++++---------
 1 file changed, 7 insertions(+), 9 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index b653582592..13fa144094 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -13,7 +13,7 @@
 
 *Author: Priya Angara*
 
-This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) <https://arxiv.org/abs/2010.06660>` and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) <https://arxiv.org/abs/1709.03489)>` for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. 
+This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) <https://arxiv.org/abs/2010.06660>`__ and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) <https://arxiv.org/abs/1709.03489)>`__ for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. 
 
 """
 
@@ -45,7 +45,7 @@
 # sequence of partial mixers :math:`U_{M, x}` which may not correspond to
 # time evolution under a fixed mixer Hamiltonian :math:`H_M`.
 #
-# .. figure:: ../demonstrations/dqva_mis/partial_mixers.png
+# .. figure:: ../demonstrations/dqva_mis/partial-mixers.png
 #    :align: center
 #    :width: 100%
 #
@@ -74,8 +74,6 @@
 # Now that we have the basics in place, let's implement the QAO-Ansatz and
 # the DQVA!
 
-#
-
 
 ######################################################################
 # Formulating the Maximum Independent Set (MIS) problem
@@ -89,7 +87,7 @@
 # MIS of a given graph. The following image shows some independent sets of
 # a graph with 5 vertices:
 #
-# .. figure:: ../demonstrations/dqva_mis/independent_sets.png
+# .. figure:: ../demonstrations/dqva_mis/independent-sets.png
 #    :align: center
 #    :width: 100%
 #
@@ -262,7 +260,7 @@ def cost_layer(gamma):
 # This extra qubit is used as an ancillary qubit to serve as the target
 # for the multi-controlled Toffoli gates.
 #
-# .. figure:: ../demonstrations/dqva_mis/mixer_unitary.png
+# .. figure:: ../demonstrations/dqva_mis/mixer-unitary.png
 #    :align: center
 #    :width: 100%
 #
@@ -490,7 +488,7 @@ def f(params):
 ######################################################################
 # Dynamic Quantum Variational Ansatz
 # ----------------------------------
-# We will now formulate the MIS using the DQVA Ansatz. The cost function
+# We will now formulate the MIS using the DQVA Ansatz [#Saleem2020]. The cost function
 # is the same as the QAO-Ansatz (the Hamming weight operator).
 #
 # In the DQVA, the way mixers are defined is slightly different from the
@@ -635,7 +633,7 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None):
 # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if
 # a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
 #
-
+#
 
 def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1):
 
@@ -790,6 +788,6 @@ def f(params):
 # References
 # ----------
 #
-# .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2000) "Approaches to Constrained Quantum Approximate Optimization",
+# .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2020) "Approaches to Constrained Quantum Approximate Optimization",
 #     `arXiv preprint arXiv:2010.06660 <https://arxiv.org/abs/2010.06660>`__.
 #

From 2429a16a556da9b4aa3ad5115785141623ad87ef Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Sun, 16 Jan 2022 10:31:30 -0800
Subject: [PATCH 09/28] fix images

---
 demonstrations/tutorial_dqva_mis.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 13fa144094..b6c6e13ca1 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -5,7 +5,7 @@
 
 .. meta::
     :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA
-    :property="og:image": https://pennylane.ai/qml/_images/mixer_unitary.png
+    :property="og:image": https://pennylane.ai/qml/_images/mixer-unitary.png
 
 .. related::
    tutorial_qaoa_intro Introduction to QAOA

From 3ce6a5605326e66649a228a99549e1d584ed8b0f Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Tue, 18 Jan 2022 10:04:51 -0800
Subject: [PATCH 10/28] fix figure size

---
 demonstrations/tutorial_dqva_mis.py   | 4 ++--
 test-results/sphinx-gallery/junit.xml | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index b6c6e13ca1..bf4febacd5 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -47,7 +47,7 @@
 #
 # .. figure:: ../demonstrations/dqva_mis/partial-mixers.png
 #    :align: center
-#    :width: 100%
+#    :width: 90%
 #
 #    ..
 #
@@ -262,7 +262,7 @@ def cost_layer(gamma):
 #
 # .. figure:: ../demonstrations/dqva_mis/mixer-unitary.png
 #    :align: center
-#    :width: 100%
+#    :width: 80%
 #
 #    ..
 #
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 7fd1edd634..2cc58ae1f5 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1 +1 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="0" time="0.0"></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="1063.1033337116241"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Solving MIS with the QAO-Ansatz and the DQVA" time="1063.1033337116241"></testcase></testsuite>
\ No newline at end of file

From c9845cc9da9dbca6a46b52305ff183477851bfd7 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Tue, 18 Jan 2022 10:42:11 -0800
Subject: [PATCH 11/28] typo

---
 demonstrations/tutorial_dqva_mis.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index bf4febacd5..9baf9e01d1 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -98,7 +98,7 @@
 # Here, independent sets can be represented as bitstrings
 # :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown
 # above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are
-# independent sets of size 0, 1, and 2 respectively. The Minimum Vertex
+# independent sets of size 0, 1, 2, and 3 respectively. The Minimum Vertex
 # Cover, i.e, vertices that cover all edges, is the dual of MIS.
 #
 # We will start with the QAO-Ansatz and then re-use most components to

From 985644c40b2f1724ff3884cac5ede5dcd751a4dc Mon Sep 17 00:00:00 2001
From: Priya Angara <pangara@uvic.ca>
Date: Wed, 19 Jan 2022 13:43:09 -0800
Subject: [PATCH 12/28] Apply suggestions from code review

Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com>
---
 demonstrations/tutorial_dqva_mis.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 9baf9e01d1..e1e82beaaa 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -31,7 +31,7 @@
 # parameters. You can learn more about this in `PennyLane's tutorial on
 # QAOA <https://pennylane.ai/qml/demos/tutorial_qaoa_intro.html>`__.
 #
-# For solving constrained combinatorial optimization problems such as MIS
+# For solving constrained combinatorial optimization problems such as maximal independent set (MIS) 
 # using QAOA, constraints can be imposed in two ways:
 #
 # 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach)
@@ -223,7 +223,7 @@ def cost_layer(gamma):
 #
 #    \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}.
 #
-# Here, where :math:`v_j` are the neighbors and :math:`l` is the number of
+# Here, where :math:`v_j` are the neighbors, :math:`l` is the number of
 # neighbors for the :math:`i`-th node and
 #
 # .. math::
@@ -774,7 +774,7 @@ def f(params):
 # mixer ordering. Furthermore, Saleem et al., recommend starting with a
 # set of initial states that are independent sets found by any classical
 # polynomial time approximation algorithm to "warm start" the
-# initialization. Furthermore, they propose the first useful application
+# initialization. They also propose the first useful application
 # of `circuit-cutting techniques <https://arxiv.org/abs/2107.07532>`__ to
 # solve MIS, which involves classical partitioning of a large graph into
 # sub-graphs, finding a solution using DQVA on a subgraph, and then

From 0a75897956c34e52a521af1835d9781fa58886e7 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Wed, 19 Jan 2022 13:43:28 -0800
Subject: [PATCH 13/28] minor edits

---
 demonstrations/tutorial_dqva_mis.py   | 4 +++-
 test-results/sphinx-gallery/junit.xml | 2 +-
 2 files changed, 4 insertions(+), 2 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 9baf9e01d1..03dc29d7f0 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -340,7 +340,7 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None):
 # function ``f`` by running the probability circuit and calculating the
 # Hamming weight (which is defined as the cost). In each iteration, an
 # ansatz is constructed based on the mixer order, with optimal parameters
-# from classical minimization of the cost function (Gradient Descent)
+# from classical minimization of the cost function using gradient descent:
 #
 # .. math::
 #
@@ -413,6 +413,8 @@ def f(params):
             probs = probability_circuit(
                 P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation
             )
+
+            # Sort bitstrings by decreasing probability
             top_counts = list(
                 map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
             )[::-1]
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 2cc58ae1f5..282bc3bbb5 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1 +1 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="1063.1033337116241"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Solving MIS with the QAO-Ansatz and the DQVA" time="1063.1033337116241"></testcase></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="488.29194808006287"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Solving MIS with the QAO-Ansatz and the DQVA" time="488.29194808006287"></testcase></testsuite>
\ No newline at end of file

From b04f721c20b87250d5f70d09f3ff153fce4ac40a Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Wed, 19 Jan 2022 13:50:26 -0800
Subject: [PATCH 14/28] add highlight to MIS

---
 demonstrations/dqva_mis/independent-sets.png | Bin 113218 -> 583857 bytes
 demonstrations/tutorial_dqva_mis.py          |   4 ++--
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/demonstrations/dqva_mis/independent-sets.png b/demonstrations/dqva_mis/independent-sets.png
index ad5457de130e700d1b703efe398add02345a7255..cff08fb023885c58871e08f32e91ffce672d8daa 100644
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diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 71f7f11b6b..501ccd783c 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -51,7 +51,7 @@
 #
 #    ..
 #
-#    Independent sets of a graph
+#    Structure of a partial mixer
 #
 # The QAO-Ansatz guarantees that the output never leaves the set of
 # feasible states (in contrast, the penalty term optimization requires an
@@ -93,7 +93,7 @@
 #
 #    ..
 #
-#    Independent sets of a graph
+#    Independent sets of a graph with the MIS highlighted.
 #
 # Here, independent sets can be represented as bitstrings
 # :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown

From d96b1f631fc95a79994bf25d5e0d604fe21fe47c Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Wed, 19 Jan 2022 13:55:29 -0800
Subject: [PATCH 15/28] apply suggestions

---
 demonstrations/dqva_mis/independent-sets.png | Bin 583857 -> 560262 bytes
 demonstrations/tutorial_dqva_mis.py          |   3 +--
 2 files changed, 1 insertion(+), 2 deletions(-)

diff --git a/demonstrations/dqva_mis/independent-sets.png b/demonstrations/dqva_mis/independent-sets.png
index cff08fb023885c58871e08f32e91ffce672d8daa..7d8d0680a8ef418b7a56bb243369f5f6076b6e23 100644
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diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 501ccd783c..b56ddd402a 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -117,11 +117,10 @@
 
 ######################################################################
 # Next, we define the graph shown above with 5 nodes for which we would
-# like to find the Maximum Independent Set.
+# like to find the Maximum Indep  endent Set.
 #
 
 edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)]
-# edges = [(0, 1), (0, 3), (0, 5), (1, 2), (1, 6), (2, 3), (2, 7), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)]
 graph = nx.Graph(edges)
 nx.draw(graph, with_labels=True)
 plt.show()

From f360c16447419405746b5ec24b37f2b33b164017 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Thu, 20 Jan 2022 14:36:01 -0800
Subject: [PATCH 16/28] change 1 -> I

---
 demonstrations/dqva_mis/independent-sets.png | Bin 560262 -> 557650 bytes
 demonstrations/tutorial_dqva_mis.py          |   4 ++--
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/demonstrations/dqva_mis/independent-sets.png b/demonstrations/dqva_mis/independent-sets.png
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diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index b56ddd402a..45f99429ac 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -152,7 +152,7 @@
 # .. math::
 #
 #
-#    b_i = \frac{1}{2} (1 - Z_i).
+#    b_i = \frac{1}{2} (I - Z_i).
 #
 # Here :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th
 # qubit.
@@ -228,7 +228,7 @@ def cost_layer(gamma):
 # .. math::
 #
 #
-#    \tilde{b}_{v_j} = \frac{1+Z_{v_j}}{2}.
+#    \tilde{b}_{v_j} = \frac{I+Z_{v_j}}{2}.
 #
 # The mixer unitary can also be written as a product of :math:`N` partial
 # mixers :math:`V_i`:

From 3ba0702ec3eb287f3c076558161621ca46b412f5 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Thu, 20 Jan 2022 14:48:49 -0800
Subject: [PATCH 17/28] add note on bitstrings

---
 demonstrations/tutorial_dqva_mis.py | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 45f99429ac..2dada3f66e 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -96,8 +96,9 @@
 #    Independent sets of a graph with the MIS highlighted.
 #
 # Here, independent sets can be represented as bitstrings
-# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. In the graph shown
-# above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are
+# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. This is convenient as the outputs of 
+# quantum computations are also bitstrings!
+# In the graph shown above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are
 # independent sets of size 0, 1, 2, and 3 respectively. The Minimum Vertex
 # Cover, i.e, vertices that cover all edges, is the dual of MIS.
 #

From 049227ec052a12b33c276a46cac6cc98fd6b526a Mon Sep 17 00:00:00 2001
From: Priya Angara <pangara@uvic.ca>
Date: Sun, 23 Jan 2022 12:03:12 -0800
Subject: [PATCH 18/28] Apply suggestions from code review

Co-authored-by: Angus Lowe <anguslowe@live.ca>
Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com>
---
 demonstrations/tutorial_dqva_mis.py | 90 +++++++++++++++--------------
 1 file changed, 47 insertions(+), 43 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 2dada3f66e..cbd643a9d4 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -18,11 +18,11 @@
 """
 
 ######################################################################
-# QAOA, QAO-Ansatz, and DQVA
+#QAO-Ansatz and DQVA
 # --------------------------
 #
 # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid
-# quantum-classical algorithm by `Farhi et
+# quantum-classical algorithm due to `Farhi et
 # al. <https://arxiv.org/abs/1411.4028>`__ is a significant meta-heuristic
 # for finding solutions to combinatorial optimization problems. The
 # quantum part involves alternating between a *cost Hamiltonian*,
@@ -37,13 +37,14 @@
 # 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach)
 # 2. By constructing the variational ansatz in a way such that constraints are satisfied at all times
 #
-# Hadfield et al., extend the QAOA framework and introduce the **Quantum
+# Hadfield et al., extend QAOA by introducing the **Quantum
 # Alternating Operator Ansatz (QAO-Ansatz)** which alternates between more
-# general families of operators thereby supporting potentially more useful
-# sets of states by focusing on unitaries rather than Hamiltonians. For
-# example, a mixing operator :math:`U_M (\beta)` is decomposed into a
-# sequence of partial mixers :math:`U_{M, x}` which may not correspond to
-# time evolution under a fixed mixer Hamiltonian :math:`H_M`.
+# general families of unitary operators. This has the potential to narrow the focus 
+# of the algorithm on a more useful set of states. For
+# example, the mixing operator :math:`U_M (\beta)` can be a product
+# of partial mixers :math:`U_{M, x}(\beta)` which may not commute. 
+# In contrast, the original proposal for QAOA used a mixing operator
+# which was a simple product of single-qubit gates.
 #
 # .. figure:: ../demonstrations/dqva_mis/partial-mixers.png
 #    :align: center
@@ -53,13 +54,13 @@
 #
 #    Structure of a partial mixer
 #
-# The QAO-Ansatz guarantees that the output never leaves the set of
-# feasible states (in contrast, the penalty term optimization requires an
-# additional pruning step since the output contains both feasible and
-# infeasible solutions). However, the QAO-Ansatz requires more complex
-# quantum circuitry, which in the case of the MIS, manifests itself as
+# The QAO-Ansatz can be used to guarantee that the state of the circuit never leaves the set of
+# feasible states. In contrast, the penalty term approach requires an
+# additional pruning step since the output can correspond to an
+# infeasible solution. However, the QAO-Ansatz requires more complicated
+# quantum circuits. In the case of MIS, one must apply
 # Multi-Controlled Toffoli gates that require high connectivity between
-# qubits. This makes the QAO-Ansatz inadequate for large graphs on current
+# qubits, limiting the practicality of the QAO-Ansatz for large graphs on 
 # near-term quantum computers.
 #
 # This brings us to the **Dynamic Quantum Variational Ansatz (DQVA)** that
@@ -118,7 +119,7 @@
 
 ######################################################################
 # Next, we define the graph shown above with 5 nodes for which we would
-# like to find the Maximum Indep  endent Set.
+# like to find the Maximum Independent Set.
 #
 
 edges = [(0, 1), (0, 3), (1, 2), (1, 3), (3, 4)]
@@ -140,7 +141,7 @@
 # For MIS, our goal is to maximize the number of vertices that form an
 # independent set, which is equivalent to maximizing the `Hamming
 # weight <https://en.wikipedia.org/wiki/Hamming_weight>`__ of the
-# bitstring. Therefore, our objective function is the Hamming weight
+# bitstring. Therefore, our objective function is encoded by the Hamming weight
 # operator,
 #
 # .. math::
@@ -186,13 +187,13 @@ def is_indset(bitstr, G):
 
 
 ######################################################################
-# Next, we define a cost unitary that incorporates the objective function,
+# Next, we define a cost unitary that incorporates the Hamming weight operator,
 # :math:`H` and is parameterized by :math:`\gamma`:
 #
 # .. math::
 #
 #
-#    U_C(\gamma) = \prod_i e^{i \gamma H}.
+#    U_C(\gamma) = e^{i \gamma H}.
 #
 
 
@@ -207,24 +208,24 @@ def cost_layer(gamma):
 # .. math::
 #
 #
-#       U_M(\beta) = \prod\_i e^{i \beta M_i}
+#       U_M(\beta) = \prod_j e^{i \beta M_j}
 #
-# where
+# where the product is over each node :math:`j`,
 #
 # .. math::
 #
 #
-#    M_i = X_i \tilde{B},
+#    M_j = X_j \tilde{B},
 #
 # and
 #
 # .. math::
 #
 #
-#    \tilde{B} = \prod_{j=1}^{l} \tilde{b}_{v_j}.
+#    \tilde{B} = \prod_{k=1}^{l} \tilde{b}_{v_k}.
 #
-# Here, where :math:`v_j` are the neighbors, :math:`l` is the number of
-# neighbors for the :math:`i`-th node and
+# Here, :math:`v_k` are the neighbors, :math:`l` is the number of
+# neighbors for the :math:`j`-th node and
 #
 # .. math::
 #
@@ -239,15 +240,14 @@ def cost_layer(gamma):
 #
 #    U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N  (I + (e^{-i\beta X_i} - I)\tilde{B}) 
 #
-# The partial mixers may not commute with each other, i.e.,
-# :math:`[V_i, V_j] \neq 0` therefore, different orderings of these mixers
-# may have different outputs. Therefore, the variational ansatz is defined
+# As mentioned earlier, the partial mixers may not commute with each other, i.e.,
+# :math:`[V_i, V_j] \neq 0`. Therefore, the variational ansatz is defined
 # up to a permutation:
 #
 # .. math::
 #
 #
-#    U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_N(\beta))
+#    U_M(\beta) \simeq \mathcal{P}(V_1(\beta)V_2(\beta)\cdots V_n(\beta))
 #
 # where P is the permutation's function of labels from :math:`1` to
 # :math:`N`.
@@ -271,8 +271,18 @@ def cost_layer(gamma):
 # In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates
 # controlled by the neighbors of the :math:`i`'th vertex.
 
-def mixer_layer(beta, ancilla, mixer_order):
-
+def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
+      """
+      Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations
+      
+      Args:
+          beta (List): A list of values for the mixing parameter
+          ancilla (int): The qubit to be used as the ancilla
+          mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+          
+      Returns: 
+          None
+      """
     # Permute the order of mixing unitaries
     if mixer_order is None:
         mixer_order = list(graph.nodes)
@@ -311,7 +321,8 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
         for qb, bit in enumerate(reversed(init_state)):
             if bit == "1":
                 qml.PauliX(wires=qb)
-    assert len(params) == 2 * P, "Incorrect number of parameters!"
+    if len(params) == 2 * P:
+      raise ValueError("Incorrect number of parameters!")
 
     betas = [a for i, a in enumerate(params) if i % 2 == 0]
     gammas = [a for i, a in enumerate(params) if i % 2 == 1]
@@ -372,9 +383,7 @@ def f(params):
         return -avg_cost
 
     # Begin outer optimization loop
-    best_indset = init_state
-    best_init_state = init_state
-    cur_init_state = init_state
+    best_indset = best_init_state = cur_init_state = init_state
     best_params = None
     best_perm = copy.copy(cur_permutation)
 
@@ -386,9 +395,7 @@ def f(params):
 
         while inner_round < 2:
             print(
-                "Start round {}.{}, Initial state = {}".format(
-                    mixer_round, inner_round, cur_init_state
-                )
+                f"Start round {mixer_round}.{inner_round}, Initial state = {cur_init_state}"
             )
 
             # Begin inner variational loop
@@ -418,7 +425,6 @@ def f(params):
             top_counts = list(
                 map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
             )[::-1]
-            # print (top_counts)
 
             best_hamming_weight = hamming_weight(best_indset)
             better_strs = []
@@ -446,9 +452,7 @@ def f(params):
             best_perm = copy.copy(cur_permutation)
             cur_init_state = best_indset
             print(
-                "\tFound new independent set: {}, Hamming weight = {}".format(
-                    best_indset, new_hamming_weight
-                )
+                f"\tFound new independent set: {best_indset}, Hamming weight = {new_hamming_weight}"
             )
 
             inner_round = inner_round + 1
@@ -466,7 +470,7 @@ def f(params):
 base_str = "0" * len(graph.nodes)
 
 out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1)
-print("Init string: {}, Best MIS: {}".format("".join(base_str), out[0]))
+print(f"Init string: {base_str}, Best MIS: {out[0]}")
 print()
 
 
@@ -483,7 +487,7 @@ def f(params):
     init_str = list(base_str)
     init_str[i] = "1"
     out = solve_mis_qaoa("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=2)
-    print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0]))
+    print(f"Init string: {init_str}, Best MIS: {out[0]}")
     print()
 
 

From be7b85de833c549b97496edd5bbc2d0b7581fbb5 Mon Sep 17 00:00:00 2001
From: Priya Angara <pangara@uvic.ca>
Date: Wed, 26 Jan 2022 11:47:23 -0800
Subject: [PATCH 19/28] Update demonstrations/tutorial_dqva_mis.py

---
 demonstrations/tutorial_dqva_mis.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index cbd643a9d4..5039790699 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -271,7 +271,7 @@ def cost_layer(gamma):
 # In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates
 # controlled by the neighbors of the :math:`i`'th vertex.
 
-def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
+def mixer_layer(beta, ancilla, mixer_order):
       """
       Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations
       

From 43b919b563001c4481af47dbcef5280dafbcc652 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Wed, 26 Jan 2022 11:49:12 -0800
Subject: [PATCH 20/28] minor edits

---
 demonstrations/tutorial_dqva_mis.py   | 45 ++++++++++++++++-----------
 test-results/sphinx-gallery/junit.xml |  6 +++-
 2 files changed, 31 insertions(+), 20 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index cbd643a9d4..bfd1785bd4 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -1,10 +1,10 @@
 r""".. _dqva_mis:
 
-Solving MIS with the QAO-Ansatz and the DQVA
-============================================
+Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization
+========================================================================
 
 .. meta::
-    :property="og:description": Solving MIS with the QAO-Ansatz and the DQVA
+    :property="og:description": Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization
     :property="og:image": https://pennylane.ai/qml/_images/mixer-unitary.png
 
 .. related::
@@ -13,7 +13,7 @@
 
 *Author: Priya Angara*
 
-This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) <https://arxiv.org/abs/2010.06660>`__ and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) <https://arxiv.org/abs/1709.03489)>`__ for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. 
+This demo discusses the `Dynamic Quantum Variational Ansatz (DQVA) <https://arxiv.org/abs/2010.06660>`__ [#Saleem2020]_ and the `Quantum Alternating Operator Ansatz (QAO-Ansatz) <https://arxiv.org/abs/1709.03489)>`__ [#Hadfield2019]_ for constrained quantum approximate optimization in the context of solving the Maximum Independent Set problem. 
 
 """
 
@@ -23,8 +23,7 @@
 #
 # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid
 # quantum-classical algorithm due to `Farhi et
-# al. <https://arxiv.org/abs/1411.4028>`__ is a significant meta-heuristic
-# for finding solutions to combinatorial optimization problems. The
+# al. <https://arxiv.org/abs/1411.4028>`__ [#Farhi2015]_  is an approach to solving combinatorial optimization problems using quantum computers. The
 # quantum part involves alternating between a *cost Hamiltonian*,
 # :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`, to evaluate the
 # objective function and a classical optimization loop updates the ansatz
@@ -272,17 +271,17 @@ def cost_layer(gamma):
 # controlled by the neighbors of the :math:`i`'th vertex.
 
 def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
-      """
-      Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations
-      
-      Args:
-          beta (List): A list of values for the mixing parameter
-          ancilla (int): The qubit to be used as the ancilla
-          mixer_order (Optional[List]=None): The desired permutation of the partial mixers
-          
-      Returns: 
-          None
-      """
+    """
+    Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations
+    
+    Args:
+        beta (List): A list of values for the mixing parameter
+        ancilla (int): The qubit to be used as the ancilla
+        mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+        
+    Returns: 
+        None
+    """
     # Permute the order of mixing unitaries
     if mixer_order is None:
         mixer_order = list(graph.nodes)
@@ -322,7 +321,7 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
             if bit == "1":
                 qml.PauliX(wires=qb)
     if len(params) == 2 * P:
-      raise ValueError("Incorrect number of parameters!")
+        raise ValueError("Incorrect number of parameters!")
 
     betas = [a for i, a in enumerate(params) if i % 2 == 0]
     gammas = [a for i, a in enumerate(params) if i % 2 == 1]
@@ -781,7 +780,7 @@ def f(params):
 # set of initial states that are independent sets found by any classical
 # polynomial time approximation algorithm to "warm start" the
 # initialization. They also propose the first useful application
-# of `circuit-cutting techniques <https://arxiv.org/abs/2107.07532>`__ to
+# of `circuit-cutting techniques <https://arxiv.org/abs/2107.07532>`__ [#Saleem2021]_ to
 # solve MIS, which involves classical partitioning of a large graph into
 # sub-graphs, finding a solution using DQVA on a subgraph, and then
 # preparing a set of states to be passed as an input, essentially
@@ -797,3 +796,11 @@ def f(params):
 # .. [#Saleem2020] Z. H. Saleem, T. Tomesh, B. Tariq, M. Suchara. (2020) "Approaches to Constrained Quantum Approximate Optimization",
 #     `arXiv preprint arXiv:2010.06660 <https://arxiv.org/abs/2010.06660>`__.
 #
+# .. [#Farhi2014] E. Farhi, J. Goldstone, S. Gutmann. (2014) "A Quantum Approximate Optimization Algorithm",
+#     `arXiv preprint arXiv:1411.4028 <https://arxiv.org/abs/1411.4028>`__.
+# 
+# .. [#Hadfield2019] S. Hadfield,  Z. Wang, B. O’Gorman, E. Rieffel, D. Venturelli, R. Biswas. (2019) "From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz",
+#     `arXiv preprint arXiv:1709.03489 <https://arxiv.org/abs/1709.03489>`__.
+#
+# .. [#Saleem2021] Z. H. Saleem, T. Tomesh, M. A. Perlin, P. Gokhale M. Suchara. (2021) "Quantum Divide and Conquer for Combinatorial Optimization and Distributed Computing",
+#     `arXiv preprint arXiv:2107.07532 <https://arxiv.org/abs/2107.07532>`__.
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 282bc3bbb5..05b2ff1dc1 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1 +1,5 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="488.29194808006287"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Solving MIS with the QAO-Ansatz and the DQVA" time="488.29194808006287"></testcase></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="1" name="sphinx-gallery" skipped="0" tests="1" time="0.6799509525299072"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="0.6799509525299072"><failure message="NameError: name 'List' is not defined">Traceback (most recent call last):
+  File "C:\Users\panga\Documents\PhD\Research\xanadu\qml\demonstrations\tutorial_dqva_mis.py", line 274, in &lt;module&gt;
+    def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
+NameError: name 'List' is not defined
+</failure></testcase></testsuite>
\ No newline at end of file

From 07a40131d3490e4d4774fd4338de643aac84260b Mon Sep 17 00:00:00 2001
From: Priya Angara <pangara@uvic.ca>
Date: Mon, 31 Jan 2022 09:58:16 -0800
Subject: [PATCH 21/28] Update demonstrations/tutorial_dqva_mis.py

Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com>
---
 demonstrations/tutorial_dqva_mis.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 72d6b56fd1..3f1783d6c8 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -65,8 +65,8 @@
 # maximizes the performance of the QAO-Ansatz by:
 #
 # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state
-# or a superposition of feasible states. A feasible state can be found by
-# using a classical approximate polynomial-time algorithm.
+#     or a superposition of feasible states. A feasible state can be found by
+#     using a classical approximate polynomial-time algorithm.
 # 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off*
 # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries
 #

From 8871c82fbcb8dbf3448415ac5ab804d1c9abac7d Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Mon, 31 Jan 2022 10:10:48 -0800
Subject: [PATCH 22/28] changes

---
 demonstrations/tutorial_dqva_mis.py   | 2 +-
 test-results/sphinx-gallery/junit.xml | 6 +-----
 2 files changed, 2 insertions(+), 6 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 72d6b56fd1..80f82ab7c7 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -633,7 +633,7 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None):
 # 1. Optimization of parameters using Gradient Descent and finding the Hamming weight with new parameters
 # 2. If the Hamming weight of the new state is larger than the initial state, the initial state gets updated to this new state
 # 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros)
-# 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved
+# 4. Steps 2 and 3 are repeated unti                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  l the Hamming weight can no longer be improved
 # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if
 # a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
 #
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 05b2ff1dc1..03ed374dee 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1,5 +1 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="1" name="sphinx-gallery" skipped="0" tests="1" time="0.6799509525299072"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="0.6799509525299072"><failure message="NameError: name 'List' is not defined">Traceback (most recent call last):
-  File "C:\Users\panga\Documents\PhD\Research\xanadu\qml\demonstrations\tutorial_dqva_mis.py", line 274, in &lt;module&gt;
-    def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
-NameError: name 'List' is not defined
-</failure></testcase></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="252.1876916885376"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="252.1876916885376"></testcase></testsuite>
\ No newline at end of file

From 06279a2b41a8cc5800eb09b7c01d76aeff664b93 Mon Sep 17 00:00:00 2001
From: Priya Angara <pangara@uvic.ca>
Date: Mon, 31 Jan 2022 12:34:13 -0800
Subject: [PATCH 23/28] Apply suggestions from code review

Co-authored-by: Angus Lowe <anguslowe@live.ca>
Co-authored-by: anthayes92 <34694788+anthayes92@users.noreply.github.com>
---
 demonstrations/tutorial_dqva_mis.py | 28 ++++++++++++++--------------
 1 file changed, 14 insertions(+), 14 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 3f1783d6c8..daca247223 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -229,15 +229,15 @@ def cost_layer(gamma):
 # .. math::
 #
 #
-#    \tilde{b}_{v_j} = \frac{I+Z_{v_j}}{2}.
+#    \tilde{b}_{v_k} = \frac{I + Z_{v_k}}{2}.
 #
 # The mixer unitary can also be written as a product of :math:`N` partial
-# mixers :math:`V_i`:
+# mixers :math:`V_j`:
 #
 # .. math::
 #
 #
-#    U_M(\beta) = \prod_{i=1}^N V_i (\beta) =\prod_{i=1}^N  (I + (e^{-i\beta X_i} - I)\tilde{B}) 
+#    U_M(\beta) = \prod_{j=1}^n V_j (\beta) =\prod_{j=1}^n  (I + (e^{-i\beta X_j} - I)\tilde{B}) 
 #
 # As mentioned earlier, the partial mixers may not commute with each other, i.e.,
 # :math:`[V_i, V_j] \neq 0`. Therefore, the variational ansatz is defined
@@ -345,11 +345,11 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None):
 
 ######################################################################
 # Finally, we run the quantum-classical loop for ``m`` different mixer
-# permutations. The expectation value is also calculated in the cost
-# function ``f`` by running the probability circuit and calculating the
-# Hamming weight (which is defined as the cost). In each iteration, an
-# ansatz is constructed based on the mixer order, with optimal parameters
-# from classical minimization of the cost function using gradient descent:
+# permutations. To evaluate the cost function ``f``, the expectation value
+# :math:`\langle C_{obj}\rangle is obtained by running the probability circuit 
+# and calculating the average Hamming weight. In each iteration, an
+# ansatz is constructed based on the mixer order and the optimal parameters
+# are determined by a classical minimization of the cost function using gradient descent.
 #
 # .. math::
 #
@@ -503,7 +503,7 @@ def f(params):
 #
 #    U_M^k(\alpha_k) = \mathcal{P}(V_1^k(\alpha_k^1)V_2^k(\alpha_k^2)\cdots V_N^k(\alpha_k^N))
 #
-# where :math:`k = 1, 2 \cdots p`.
+# where :math:`k = 1, 2, \dots, p`.
 #
 # In the DQVA, whenever the :math:`j`-th bit of the initial state is one,
 # the corresponding parameter :math:`\alpha_k^j` is set to 0. For example,
@@ -585,9 +585,9 @@ def dqva_ansatz(P, params=[], init_state=None, mixer_order=None):
                 qml.PauliX(wires=qb)
 
     num_nonzero = nq - hamming_weight(init_state)
-    assert len(params) == (nq + 1) * P, "Incorrect number of parameters!"
-    alpha_list = []
-    gamma_list = []
+    if len(params) == (nq + 1) * P:
+        raise ValueError("Incorrect number of parameters!")
+    alpha_list = gamma_list = []
     last_idx = 0
     for p in range(P):
         chunk = num_nonzero + 1
@@ -635,7 +635,7 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None):
 # 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros)
 # 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved
 # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if
-# a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
+#    a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
 #
 #
 
@@ -755,7 +755,7 @@ def f(params):
     init_str = list(base_str)
     init_str[i] = "1"
     out = solve_mis_dqva("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=1)
-    print("Init string: {}, Best MIS: {}".format("".join(init_str), out[0]))
+    print(f"Init string: {init_str}, Best MIS: {out[0]}")
     print()
 
 

From 32119b1c897f6ff0a44562a74ca19a0c64ae9f0d Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Mon, 31 Jan 2022 12:51:21 -0800
Subject: [PATCH 24/28] changes

---
 demonstrations/tutorial_dqva_mis.py | 37 ++++++++++++++++-------------
 1 file changed, 20 insertions(+), 17 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 3c8d85c2f6..8ebeed9822 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -65,8 +65,8 @@
 # maximizes the performance of the QAO-Ansatz by:
 #
 # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state
-#     or a superposition of feasible states. A feasible state can be found by
-#     using a classical approximate polynomial-time algorithm.
+# or a superposition of feasible states. A feasible state can be found by
+# using a classical approximate polynomial-time algorithm.
 # 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off*
 # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries
 #
@@ -95,11 +95,10 @@
 #    Independent sets of a graph with the MIS highlighted.
 #
 # Here, independent sets can be represented as bitstrings
-# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`. This is convenient as the outputs of 
+# :math:`b = \{b_1, b_2, \cdots b_n\} \in \{0, 1\}^n`, where the i'th bit being one represents inclusion of the ith vertex in the set. This is convenient as the outputs of 
 # quantum computations are also bitstrings!
 # In the graph shown above :math:`00000`, :math:`10000`, :math:`01001`, :math:`10101` are
-# independent sets of size 0, 1, 2, and 3 respectively. The Minimum Vertex
-# Cover, i.e, vertices that cover all edges, is the dual of MIS.
+# independent sets of size 0, 1, 2, and 3 respectively.
 #
 # We will start with the QAO-Ansatz and then re-use most components to
 # formulate the DQVA.
@@ -171,20 +170,24 @@ def hamming_weight(bitstr):
     return sum([1 for bit in bitstr if bit == "1"])
 
 
+# def is_indset(bitstr, G):
+#     nodes = list(G.nodes)
+#     ind_set = []
+#     for idx, bit in enumerate(bitstr):
+#         if bit == "1":
+#             cur_neighbors = list(G.neighbors(idx))
+#             for node in ind_set:
+#                 if node in cur_neighbors:
+#                     return False
+#             else:
+#                 ind_set.append(idx)
+#     return True
+
 def is_indset(bitstr, G):
-    nodes = list(G.nodes)
-    ind_set = []
-    for idx, bit in enumerate(bitstr):
-        if bit == "1":
-            cur_neighbors = list(G.neighbors(idx))
-            for node in ind_set:
-                if node in cur_neighbors:
-                    return False
-            else:
-                ind_set.append(idx)
+    for edge in list(G.edges):
+        if bitstr[edge[0]] == "1" and bitstr[edge[1]] == "1":
+            return False
     return True
-
-
 ######################################################################
 # Next, we define a cost unitary that incorporates the Hamming weight operator,
 # :math:`H` and is parameterized by :math:`\gamma`:

From 91988e70697d0af42f7a9f28bb26e778db01e07f Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Thu, 3 Feb 2022 22:27:50 -0800
Subject: [PATCH 25/28] apply suggestions

---
 demonstrations/tutorial_dqva_mis.py   | 258 ++++++++++++--------------
 test-results/sphinx-gallery/junit.xml |   2 +-
 2 files changed, 123 insertions(+), 137 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index b7da11065d..9bef9848b6 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -23,8 +23,8 @@
 #
 # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid
 # quantum-classical technique due to `Farhi et
-# al. <https://arxiv.org/abs/1411.4028>`__ [#Farhi2015]_  is an approach to solving combinatorial optimization problems using quantum computers. The
-# quantum part evaluates the objective function and involves alternating between a *cost Hamiltonian*,
+# al. <https://arxiv.org/abs/1411.4028>`__ [#Farhi2014]_  is an approach to solving combinatorial optimization problems using quantum computers. The
+# quantum part evaluates the objective function and involves alternating between unitaries corresponding to a *cost Hamiltonian*,
 # :math:`H_C`, and a *mixer Hamiltonian*, :math:`H_M`. A classical optimization loop updates the ansatz
 # parameters. You can learn more about this in `PennyLane's tutorial on
 # QAOA <https://pennylane.ai/qml/demos/tutorial_qaoa_intro.html>`__.
@@ -65,8 +65,8 @@
 # maximizes the performance of the QAO-Ansatz by:
 #
 # 1. *Warm starting the optimization* by starting with an initial state that is a feasible state
-# or a superposition of feasible states. A feasible state can be found by
-# using a classical approximate polynomial-time algorithm.
+#     or a superposition of feasible states. A feasible state can be found by
+#     using a classical approximate polynomial-time algorithm.
 # 2. *Updating the ansatz dynamically* by turning partial mixers *on* and *off*
 # 3. *Randomizing the ordering of the partial mixers* in the mixer unitaries
 #
@@ -112,8 +112,7 @@
 import networkx as nx
 from matplotlib import pyplot as plt
 import copy
-from typing import List, Optional
-
+from typing import List, Optional, Tuple, Callable
 
 ######################################################################
 # Next, we define the graph shown above with 5 nodes for which we would
@@ -134,7 +133,6 @@
 wires = len(graph.nodes) + 1
 dev = qml.device("default.qubit", wires=range(wires))
 
-
 ######################################################################
 # For MIS, our goal is to maximize the number of vertices that form an
 # independent set, which is equivalent to maximizing the `Hamming
@@ -166,28 +164,17 @@
 #
 
 
-def hamming_weight(bitstr):
+def hamming_weight(bitstr: str) -> int:
     return sum([1 for bit in bitstr if bit == "1"])
 
 
-# def is_indset(bitstr, G):
-#     nodes = list(G.nodes)
-#     ind_set = []
-#     for idx, bit in enumerate(bitstr):
-#         if bit == "1":
-#             cur_neighbors = list(G.neighbors(idx))
-#             for node in ind_set:
-#                 if node in cur_neighbors:
-#                     return False
-#             else:
-#                 ind_set.append(idx)
-#     return True
-
-def is_indset(bitstr, G):
+def is_indset(bitstr:str, G: nx.Graph) -> bool:
     for edge in list(G.edges):
         if bitstr[edge[0]] == "1" and bitstr[edge[1]] == "1":
             return False
     return True
+
+
 ######################################################################
 # Next, we define a cost unitary that incorporates the Hamming weight operator,
 # :math:`H` and is parameterized by :math:`\gamma`:
@@ -199,7 +186,7 @@ def is_indset(bitstr, G):
 #
 
 
-def cost_layer(gamma):
+def cost_layer(gamma: List) -> None:
     for qb in graph.nodes:
         qml.RZ(2 * gamma, wires=qb)
 
@@ -264,16 +251,41 @@ def cost_layer(gamma):
 #
 # .. figure:: ../demonstrations/dqva_mis/mixer-unitary.png
 #    :align: center
-#    :width: 80%
+#    :width: 60%
 #
 #    ..
 #
 #    Formulation of the mixer unitary
 #
 # In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates
-# controlled by the neighbors of the :math:`i`'th vertex.
+# controlled by the neighbors of the :math:`i`'th vertex. Let us define a 
+# helper function that applies the operations shown in the figure above. 
+
+
+def apply_multi_controlled_toffoli(ancilla: int, beta:List, qubit: int) -> None:
+    """Helper function to apply the multi-controlled Toffoli, targeting the ancilla qubit"""
+
+    neighbors = list(graph.neighbors(qubit))
+
+    # Apply the multi-controlled Toffoli, targetting the ancilla qubit
+    ctrl_qubits = [i for i in neighbors]
+
+    qml.MultiControlledX(
+        control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
+    )
 
-def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
+    qml.CRX(2 * beta, wires=[ancilla, qubit])
+
+    # Uncompute the ancilla
+    qml.MultiControlledX(
+        control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
+    )
+
+# We are now ready to build the mixer layer for the QAO-Ansatz by applying the
+# multi-controlled Toffoli operations on each of the qubits in the graph. 
+
+
+def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None) -> None:
     """
     Builds the QAO-Ansatz mixer layer for max independent set problem using PennyLane operations
     
@@ -290,31 +302,20 @@ def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None):
         mixer_order = list(graph.nodes)
 
     for qubit in list(mixer_order):
-        neighbors = list(graph.neighbors(qubit))
-
-        # Apply the multi-controlled Toffoli, targetting the ancilla qubit
-        ctrl_qubits = [i for i in neighbors]
-        qml.MultiControlledX(
-            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
-        )
-
-        qml.CRX(2 * beta, wires=[ancilla, qubit])
-
-        # Uncompute the ancilla
-        qml.MultiControlledX(
-            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
-        )
+        apply_multi_controlled_toffoli(ancilla, beta, qubit)
 
 ######################################################################
 # We are now ready to build the circuit consisting of alternating cost and
 # mixer layers. A list of ``params`` includes parameters for the mixer
 # layer (even elements) and cost layer (odd elements).
-#
-
+# First let's define a helper function to initialize the ansatz. 
+# This function applies Pauli-X operations on qubits to   
+# prepare an initial state.
 
-def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
+def initialize_ansatz(init_state: Optional[str]=None) -> None:
+    """ Helper function to initialize circuit ansatz"""
     nq = len(graph.nodes)
-    # Initialize
+
     if init_state is None:
         init_state = "0" * nq
 
@@ -322,6 +323,13 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
         for qb, bit in enumerate(reversed(init_state)):
             if bit == "1":
                 qml.PauliX(wires=qb)
+
+
+def qaoa_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None:
+    nq = len(graph.nodes)
+    
+    initialize_ansatz(init_state)
+    
     if len(params) != 2 * P:
         raise ValueError("Incorrect number of parameters!")
 
@@ -334,6 +342,7 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
         cost_layer(gamma)
 
 
+
 ######################################################################
 # Now, let's build the circuit that finds the probabilities of measuring
 # each bitstring.
@@ -341,7 +350,7 @@ def qaoa_ansatz(P, params=[], init_state=None, mixer_order=None):
 
 
 @qml.qnode(dev)
-def probability_circuit(P, params=[], init_state=None, mixer_order=None):
+def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order:  Optional[List]=None) -> np.ndarray:
     qaoa_ansatz(P, params, init_state, mixer_order)
     return qml.probs(wires=range(wires - 1))
 
@@ -352,16 +361,50 @@ def probability_circuit(P, params=[], init_state=None, mixer_order=None):
 # :math:`\langle C_{obj}\rangle is obtained by running the probability circuit 
 # and calculating the average Hamming weight. In each iteration, an
 # ansatz is constructed based on the mixer order and the optimal parameters
-# are determined by a classical minimization of the cost function using gradient descent.
-#
-# .. math::
-#
-#
-#    \langle \beta, \gamma \vert C_{obj} \vert \beta, \gamma \rangle.
+# are determined by a classical minimization of the cost function.
 #
+# Let us construct the optimizer that uses gradient descent to minimize a 
+# cost function f. 
+
+def optimize_params(f: Callable[[List], float], init_params: List) -> Tuple[List, float]:
+    """ Helper function to optimize circuit parameters"""
+
+    optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
+    cur_params = init_params.copy()
+
+    for i in range(70):
+        cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
+
+    return cur_params, opt_cost
+
 
+# We will also define a helper function that compares the independent sets 
+# generated by the solver with a currently-known best independent set.
+# This is done by comparing the hamming weights of the bitstrings 
+# with the highest probabilities with the best independent set. 
 
-def solve_mis_qaoa(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1):
+
+def better_ind_sets(probs: np.ndarray, best_indset, cutoff) -> List:
+    """ Helper function to calculate better independent sets"""
+    top_counts = list(
+        map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
+    )[::-1]
+
+    best_hamming_weight = hamming_weight(best_indset)
+    better_strs = []
+
+    print(top_counts[:cutoff])
+    for bitstr in top_counts[:cutoff]:
+
+        this_hamming = hamming_weight(bitstr)
+        if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
+            better_strs.append((bitstr, this_hamming))
+
+    better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
+    return better_strs  
+
+def solve_mis_qaoa(init_state: str, P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, 
+                   threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]:
     # Select an ordering for the partial mixers
     if mixer_order == None:
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
@@ -405,15 +448,9 @@ def f(params):
 
             init_params = np.random.uniform(low=-np.pi, high=np.pi, size=num_params)
             print("\tCurrent Mixer Order:", cur_permutation)
-
+            
             # Optimize parameters
-            optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
-            cur_params = init_params.copy()
-
-            for i in range(70):
-                cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
-
-            opt_params = cur_params.copy()
+            opt_params, opt_cost = optimize_params(f, init_params)
 
             print("\tOptimal cost:", opt_cost)
 
@@ -423,21 +460,7 @@ def f(params):
             )
 
             # Sort bitstrings by decreasing probability
-            top_counts = list(
-                map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
-            )[::-1]
-
-            best_hamming_weight = hamming_weight(best_indset)
-            better_strs = []
-
-            print(top_counts[:cutoff])
-            for bitstr in top_counts[:cutoff]:
-
-                this_hamming = hamming_weight(bitstr)
-                if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
-                    better_strs.append((bitstr, this_hamming))
-
-            better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
+            better_strs = better_ind_sets(probs, best_indset, cutoff)
 
             # If no improvement was made, break and go to next mixer round
             if len(better_strs) == 0:
@@ -464,6 +487,7 @@ def f(params):
     return best_indset, best_params, best_init_state, best_perm
 
 
+
 ######################################################################
 # Let's run this to find the MIS!
 #
@@ -471,10 +495,9 @@ def f(params):
 base_str = "0" * len(graph.nodes)
 
 out = solve_mis_qaoa(base_str, P=1, m=4, threshold=1e-5, cutoff=1)
-print(f"Init string: {base_str}, Best MIS: {out[0]}")
+print(f"Init string: {base_str}, Best MIS: {out}")
 print()
 
-
 ######################################################################
 # Starting with an all-zero initial string will give us an independent
 # set, but this may not be the maximum. Now, we run this with a few
@@ -487,15 +510,15 @@ def f(params):
 for i in range(len(graph.nodes)):
     init_str = list(base_str)
     init_str[i] = "1"
-    out = solve_mis_qaoa("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=2)
-    print(f"Init string: {init_str}, Best MIS: {out[0]}")
+    input_string = "".join(init_str)
+    out = solve_mis_qaoa(input_string, P=1, m=4, threshold=1e-5, cutoff=2)
+    print(f"Init string: {input_string}, Best MIS: {out[0]}")
     print()
 
-
 ######################################################################
 # Dynamic Quantum Variational Ansatz
 # ----------------------------------
-# We will now formulate the MIS using the DQVA Ansatz [#Saleem2020]. The cost function
+# We will now formulate the MIS using the DQVA Ansatz [#Saleem2020]_. The cost function
 # is the same as the QAO-Ansatz (the Hamming weight operator).
 #
 # In the DQVA, the way mixers are defined is slightly different from the
@@ -522,7 +545,7 @@ def f(params):
 #
 
 
-def mixer_dqva(alpha, ancilla, init_state, mixer_order):
+def mixer_dqva(alpha: List, ancilla: int, init_state: str, mixer_order: Optional[List]=None) -> None:
 
     # Permute the order of mixing unitaries
     if mixer_order is None:
@@ -544,21 +567,7 @@ def mixer_dqva(alpha, ancilla, init_state, mixer_order):
         if pad_alpha[qubit] == None or not graph.has_node(qubit):
             # Turn off mixers for qubits which are already 1
             continue
-
-        neighbors = list(graph.neighbors(qubit))
-
-        # Apply the multi-controlled Toffoli, targetting the ancilla qubit
-        ctrl_qubits = [i for i in neighbors]
-        qml.MultiControlledX(
-            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
-        )
-
-        qml.CRX(2 * pad_alpha[qubit], wires=[ancilla, qubit])
-
-        # Uncompute the ancilla
-        qml.MultiControlledX(
-            control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
-        )
+        apply_multi_controlled_toffoli(ancilla, pad_alpha[qubit], qubit)
 
 ######################################################################
 # The structure of the cost and mixer unitaries is similar QAO-Ansatz,
@@ -575,22 +584,16 @@ def mixer_dqva(alpha, ancilla, init_state, mixer_order):
 #
 
 
-def dqva_ansatz(P, params=[], init_state=None, mixer_order=None):
+def dqva_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None:
     nq = len(graph.nodes)
 
-    # Step1: Initialize
-    if init_state is None:
-        init_state = "0" * nq
-
-    else:
-        for qb, bit in enumerate(reversed(init_state)):
-            if bit == "1":
-                qml.PauliX(wires=qb)
+    initialize_ansatz(init_state)
 
     num_nonzero = nq - hamming_weight(init_state)
-    if len(params) == (nq + 1) * P:
+    if len(params) != (nq + 1) * P:
         raise ValueError("Incorrect number of parameters!")
-    alpha_list = gamma_list = []
+    alpha_list = []
+    gamma_list = []
     last_idx = 0
     for p in range(P):
         chunk = num_nonzero + 1
@@ -612,13 +615,14 @@ def dqva_ansatz(P, params=[], init_state=None, mixer_order=None):
 
 
 
+
 ######################################################################
 # Let's also define the probability circuit for the DQVA ansatz
 #
 
 
 @qml.qnode(dev)
-def probability_dqva(P, params=[], init_state=None, mixer_order=None):
+def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> np.ndarray:
     dqva_ansatz(P, params, init_state, mixer_order)
     return qml.probs(wires=dev.wires[:-1])
 
@@ -636,13 +640,13 @@ def probability_dqva(P, params=[], init_state=None, mixer_order=None):
 # 1. Optimization of parameters using Gradient Descent and finding the Hamming weight with new parameters
 # 2. If the Hamming weight of the new state is larger than the initial state, the initial state gets updated to this new state
 # 3. Based on this new state, partial mixers are updated (i.e., turned off for ones and turned on for zeros)
-# 4. Steps 2 and 3 are repeated unti                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  l the Hamming weight can no longer be improved
+# 4. Steps 2 and 3 are repeated until the Hamming weight can no longer be improved
 # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if
 #    a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
 #
 #
 
-def solve_mis_dqva(init_state, P=1, m=1, mixer_order=None, threshold=1e-5, cutoff=1):
+def solve_mis_dqva(init_state: Optional[str], P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]:
 
     # Select an ordering for the partial mixers
     if mixer_order == None:
@@ -665,9 +669,7 @@ def f(params):
         return -avg_cost
 
     # Begin outer optimization loop
-    best_indset = init_state
-    best_init_state = init_state
-    cur_init_state = init_state
+    best_indset = best_init_state = cur_init_state = init_state
     best_params = None
     best_perm = copy.copy(cur_permutation)
 
@@ -691,13 +693,7 @@ def f(params):
             print("\tCurrent Mixer Order:", cur_permutation)
 
             # Optimize parameters
-            optimizer = qml.GradientDescentOptimizer(stepsize=0.5)
-            cur_params = init_params.copy()
-
-            for i in range(70):
-                cur_params, opt_cost = optimizer.step_and_cost(f, cur_params)
-
-            opt_params = cur_params.copy()
+            opt_params, opt_cost = optimize_params(f, init_params)
 
             print("\tOptimal cost:", opt_cost)
 
@@ -705,18 +701,8 @@ def f(params):
                 P, params=opt_params, init_state=cur_init_state, mixer_order=cur_permutation
             )
 
-            top_counts = list(
-                map(lambda x: np.binary_repr(x, len(graph.nodes)), np.argsort(probs))
-            )[::-1]
-
-            best_hamming_weight = hamming_weight(best_indset)
-            better_strs = []
-
-            for bitstr in top_counts[:1]:
-                this_hamming = hamming_weight(bitstr)
-                if is_indset(bitstr, graph) and this_hamming > best_hamming_weight:
-                    better_strs.append((bitstr, this_hamming))
-            better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
+            # Sort bitstrings by decreasing probability
+            better_strs = better_ind_sets(probs, best_indset, cutoff)
 
             # If no improvement was made, break and go to next mixer round
             if len(better_strs) == 0:
@@ -753,12 +739,12 @@ def f(params):
 #
 
 base_str = "0" * len(graph.nodes)
-all_init_strs = []
 for i in range(len(graph.nodes)):
     init_str = list(base_str)
     init_str[i] = "1"
-    out = solve_mis_dqva("".join(init_str), P=1, m=4, threshold=1e-5, cutoff=1)
-    print(f"Init string: {init_str}, Best MIS: {out[0]}")
+    input_string = "".join(init_str)
+    out = solve_mis_dqva(input_string, P=1, m=4, threshold=1e-5, cutoff=1)
+    print(f"Init string: {input_string}, Best MIS: {out[0]}")
     print()
 
 
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 03ed374dee..24f7a4ec36 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1 +1 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="252.1876916885376"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="252.1876916885376"></testcase></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="214.2583453655243"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="214.2583453655243"></testcase></testsuite>
\ No newline at end of file

From 167d23ae6cb77384faac2d14be1ea99c11616ea8 Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Thu, 3 Feb 2022 22:29:25 -0800
Subject: [PATCH 26/28] apply suggestions

---
 demonstrations/tutorial_dqva_mis.py | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 9bef9848b6..7c18c47aa3 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -312,6 +312,7 @@ def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None) -> N
 # This function applies Pauli-X operations on qubits to   
 # prepare an initial state.
 
+
 def initialize_ansatz(init_state: Optional[str]=None) -> None:
     """ Helper function to initialize circuit ansatz"""
     nq = len(graph.nodes)
@@ -366,6 +367,7 @@ def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[
 # Let us construct the optimizer that uses gradient descent to minimize a 
 # cost function f. 
 
+
 def optimize_params(f: Callable[[List], float], init_params: List) -> Tuple[List, float]:
     """ Helper function to optimize circuit parameters"""
 
@@ -403,6 +405,7 @@ def better_ind_sets(probs: np.ndarray, best_indset, cutoff) -> List:
     better_strs = sorted(better_strs, key=lambda t: t[1], reverse=True)
     return better_strs  
 
+
 def solve_mis_qaoa(init_state: str, P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, 
                    threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]:
     # Select an ordering for the partial mixers
@@ -644,7 +647,7 @@ def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str
 # 5. If no new Hamming weight is obtained, the partial mixers are randomized and steps 2 and 3 are repeated to check if
 #    a better Hamming weight is found. The number of randomizations is controlled via a hyperparameter.
 #
-#
+
 
 def solve_mis_dqva(init_state: Optional[str], P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]:
 

From 14b48e29a8fabe11025941f9769f2c1bcb8d51aa Mon Sep 17 00:00:00 2001
From: pangara <pangara@uvic.ca>
Date: Fri, 4 Feb 2022 10:14:01 -0800
Subject: [PATCH 27/28] add docstrings

---
 demonstrations/tutorial_dqva_mis.py   | 87 ++++++++++++++++++++++++++-
 test-results/sphinx-gallery/junit.xml |  2 +-
 2 files changed, 85 insertions(+), 4 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 7c18c47aa3..29940e8658 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -165,10 +165,13 @@
 
 
 def hamming_weight(bitstr: str) -> int:
+    """Helper function to calculate the hamming weight of a bitstring"""
     return sum([1 for bit in bitstr if bit == "1"])
 
 
 def is_indset(bitstr:str, G: nx.Graph) -> bool:
+    """Helper function that returns True when a bitstring is a valid independent set"""
+
     for edge in list(G.edges):
         if bitstr[edge[0]] == "1" and bitstr[edge[1]] == "1":
             return False
@@ -187,6 +190,15 @@ def is_indset(bitstr:str, G: nx.Graph) -> bool:
 
 
 def cost_layer(gamma: List) -> None:
+    """
+    Builds the QAO-Ansatz cost layer for max independent set problem using PennyLane operations
+    
+    Args:
+        gamma (List): A list of values for the cost parameter
+        
+    Returns: 
+        None
+    """
     for qb in graph.nodes:
         qml.RZ(2 * gamma, wires=qb)
 
@@ -314,7 +326,7 @@ def mixer_layer(beta: List, ancilla: int, mixer_order: Optional[List]=None) -> N
 
 
 def initialize_ansatz(init_state: Optional[str]=None) -> None:
-    """ Helper function to initialize circuit ansatz"""
+    """Helper function to initialize circuit ansatz"""
     nq = len(graph.nodes)
 
     if init_state is None:
@@ -327,6 +339,18 @@ def initialize_ansatz(init_state: Optional[str]=None) -> None:
 
 
 def qaoa_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None:
+    """
+    Builds the QAO-Ansatz for max independent set problem
+    
+    Args:
+        P (int): Number of parameters
+        params (Optional[List]=[]): A list of 2*P parameters
+        init_state (Optional[str]=None): Bitstring representing the initial state
+        mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+        
+    Returns: 
+        None
+    """
     nq = len(graph.nodes)
     
     initialize_ansatz(init_state)
@@ -352,6 +376,8 @@ def qaoa_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=Non
 
 @qml.qnode(dev)
 def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order:  Optional[List]=None) -> np.ndarray:
+    """Obtains probabilities for the QAO-Ansatz given a set of parameters, initial state, and the mixer order"""
+
     qaoa_ansatz(P, params, init_state, mixer_order)
     return qml.probs(wires=range(wires - 1))
 
@@ -408,6 +434,22 @@ def better_ind_sets(probs: np.ndarray, best_indset, cutoff) -> List:
 
 def solve_mis_qaoa(init_state: str, P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, 
                    threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]:
+    """
+    Solver for max independent set problem using the QAO-Ansatz 
+    
+    Args:
+        init_state (Optional[str]=None): Bitstring representing the initial state
+        P (int): Number of parameters
+        m (int): Number of mixer rounds
+        mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+        threshold (Optional[float]=1e-5): A threshold to remove low probability outcomes 
+        cutoff (Optional[int]=1): number of "better" independent sets to consider
+    Returns: 
+        str: Best independent set
+        np.ndarray: Best parameters
+        str: Best initial state
+        List: Best mixer order
+    """
     # Select an ordering for the partial mixers
     if mixer_order == None:
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
@@ -549,7 +591,17 @@ def f(params):
 
 
 def mixer_dqva(alpha: List, ancilla: int, init_state: str, mixer_order: Optional[List]=None) -> None:
-
+    """
+    Builds the DQVA mixer layer for max independent set problem using PennyLane operations
+    
+    Args:
+        beta (List): A list of values for the mixing parameter
+        ancilla (int): The qubit to be used as the ancilla
+        mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+        
+    Returns: 
+        None
+    """
     # Permute the order of mixing unitaries
     if mixer_order is None:
         mixer_order = list(graph.nodes)
@@ -588,6 +640,18 @@ def mixer_dqva(alpha: List, ancilla: int, init_state: str, mixer_order: Optional
 
 
 def dqva_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> None:
+    """
+    Builds the DQVA for max independent set problem
+    
+    Args:
+        P (int): Number of parameters
+        params (Optional[List]=[]): a list of 2*P parameters
+        init_state (Optional[str]=None): bitstring representing the initial state
+        mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+        
+    Returns: 
+        None
+    """
     nq = len(graph.nodes)
 
     initialize_ansatz(init_state)
@@ -626,6 +690,8 @@ def dqva_ansatz(P: int, params: Optional[List]=[], init_state: Optional[str]=Non
 
 @qml.qnode(dev)
 def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str]=None, mixer_order: Optional[List]=None) -> np.ndarray:
+    """Obtains probabilities for the DQVA given a set of parameters, initial state, and the mixer order"""
+
     dqva_ansatz(P, params, init_state, mixer_order)
     return qml.probs(wires=dev.wires[:-1])
 
@@ -650,7 +716,22 @@ def probability_dqva(P: int, params: Optional[List]=[], init_state: Optional[str
 
 
 def solve_mis_dqva(init_state: Optional[str], P: Optional[int]=1, m: Optional[int]=1, mixer_order: Optional[List]=None, threshold: Optional[float]=1e-5, cutoff: Optional[int]=1) -> Tuple[str, np.ndarray, str, List]:
-
+    """
+    Solver for max independent set problem using the DQVA 
+    
+    Args:
+        init_state (Optional[str]=None): Bitstring representing the initial state
+        P (int): Number of parameters
+        m (int): Number of mixer rounds
+        mixer_order (Optional[List]=None): The desired permutation of the partial mixers
+        threshold (Optional[float]=1e-5): A threshold to remove low probability outcomes 
+        cutoff (Optional[int]=1): number of "better" independent sets to consider
+    Returns: 
+        str: Best independent set
+        np.ndarray: Best parameters
+        str: Best initial state
+        List: Best mixer order
+    """
     # Select an ordering for the partial mixers
     if mixer_order == None:
         cur_permutation = np.random.permutation(list(graph.nodes)).tolist()
diff --git a/test-results/sphinx-gallery/junit.xml b/test-results/sphinx-gallery/junit.xml
index 24f7a4ec36..c3f661e6b4 100644
--- a/test-results/sphinx-gallery/junit.xml
+++ b/test-results/sphinx-gallery/junit.xml
@@ -1 +1 @@
-<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="214.2583453655243"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="214.2583453655243"></testcase></testsuite>
\ No newline at end of file
+<?xml version="1.0" encoding="utf-8"?><testsuite errors="0" failures="0" name="sphinx-gallery" skipped="0" tests="1" time="951.0791280269623"><testcase classname="tutorial_dqva_mis" file="demonstrations\tutorial_dqva_mis.py" line="1" name="Dynamic Quantum Variational Ansatz (DQVA) for Combinatorial Optimization" time="951.0791280269623"></testcase></testsuite>
\ No newline at end of file

From c3e4f7fb4b306c7c432ef7833068863bc8af9300 Mon Sep 17 00:00:00 2001
From: Priya Angara <pangara@uvic.ca>
Date: Tue, 8 Feb 2022 09:25:12 -0800
Subject: [PATCH 28/28] Apply suggestions from code review

Co-authored-by: Olivia Di Matteo <2068515+glassnotes@users.noreply.github.com>
---
 demonstrations/tutorial_dqva_mis.py | 23 +++++++++++++----------
 1 file changed, 13 insertions(+), 10 deletions(-)

diff --git a/demonstrations/tutorial_dqva_mis.py b/demonstrations/tutorial_dqva_mis.py
index 29940e8658..340685dced 100644
--- a/demonstrations/tutorial_dqva_mis.py
+++ b/demonstrations/tutorial_dqva_mis.py
@@ -18,7 +18,7 @@
 """
 
 ######################################################################
-#QAO-Ansatz and DQVA
+# QAO-Ansatz and DQVA
 # --------------------------
 #
 # The **Quantum Approximate Optimization Algorithm (QAOA)**, a hybrid
@@ -29,7 +29,7 @@
 # parameters. You can learn more about this in `PennyLane's tutorial on
 # QAOA <https://pennylane.ai/qml/demos/tutorial_qaoa_intro.html>`__.
 #
-# For solving constrained combinatorial optimization problems such as maximal independent set (MIS) 
+# For solving constrained combinatorial optimization problems such as maximum independent set (MIS) 
 # using QAOA, constraints can be imposed in two ways:
 #
 # 1. By adding a penalty term to the problem's objective function (the Lagrange multiplier approach)
@@ -57,7 +57,7 @@
 # additional pruning step since the output can correspond to an
 # infeasible solution. However, the QAO-Ansatz requires more complicated
 # quantum circuits. In the case of MIS, one must apply
-# Multi-Controlled Toffoli gates that require high connectivity between
+# multi-controlled Toffoli gates that require high connectivity between
 # qubits, limiting the practicality of the QAO-Ansatz for large graphs on 
 # near-term quantum computers.
 #
@@ -78,7 +78,7 @@
 # Formulating the Maximum Independent Set (MIS) problem
 # -----------------------------------------------------
 #
-# Given a graph, :math:`G = (V, E)` where :math:`V` is the set of vertices
+# Given a graph :math:`G = (V, E)` where :math:`V` is the set of vertices
 # and :math:`E` is the set of edges, an *independent set* is a subset of
 # vertices, :math:`V' \subset V`, such that no two vertices in :math:`V'`
 # share an edge. A *Maximum Independent Set* (MIS) is an independent set of
@@ -159,7 +159,7 @@
 
 ######################################################################
 # We need a couple of helper functions here: ``hamming_weight``, which
-# calculates the Hamming weight of a given bitstring and ``is_indset``,
+# calculates the Hamming weight of a given bitstring, and ``is_indset``,
 # which verifies whether a given bitstring is a valid independent set.
 #
 
@@ -180,7 +180,7 @@ def is_indset(bitstr:str, G: nx.Graph) -> bool:
 
 ######################################################################
 # Next, we define a cost unitary that incorporates the Hamming weight operator,
-# :math:`H` and is parameterized by :math:`\gamma`:
+# :math:`H`, and is parameterized by :math:`\gamma`:
 #
 # .. math::
 #
@@ -269,7 +269,7 @@ def cost_layer(gamma: List) -> None:
 #
 #    Formulation of the mixer unitary
 #
-# In the figure above, a PauliX rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates
+# In the figure above, an :math:`X` rotation is applied to qubit :math:`\vert i \rangle` between two multi-controlled Toffoli gates
 # controlled by the neighbors of the :math:`i`'th vertex. Let us define a 
 # helper function that applies the operations shown in the figure above. 
 
@@ -293,6 +293,7 @@ def apply_multi_controlled_toffoli(ancilla: int, beta:List, qubit: int) -> None:
         control_wires=ctrl_qubits, wires=[ancilla], control_values="0" * len(neighbors)
     )
 
+######################################################################
 # We are now ready to build the mixer layer for the QAO-Ansatz by applying the
 # multi-controlled Toffoli operations on each of the qubits in the graph. 
 
@@ -385,7 +386,7 @@ def probability_circuit(P: int, params: Optional[List]=[], init_state: Optional[
 ######################################################################
 # Finally, we run the quantum-classical loop for ``m`` different mixer
 # permutations. To evaluate the cost function ``f``, the expectation value
-# :math:`\langle C_{obj}\rangle is obtained by running the probability circuit 
+# :math:`\langle C_{obj}\rangle` is obtained by running the probability circuit 
 # and calculating the average Hamming weight. In each iteration, an
 # ansatz is constructed based on the mixer order and the optimal parameters
 # are determined by a classical minimization of the cost function.
@@ -406,9 +407,11 @@ def optimize_params(f: Callable[[List], float], init_params: List) -> Tuple[List
     return cur_params, opt_cost
 
 
+
+######################################################################
 # We will also define a helper function that compares the independent sets 
 # generated by the solver with a currently-known best independent set.
-# This is done by comparing the hamming weights of the bitstrings 
+# This is done by comparing the Hamming weights of the bitstrings 
 # with the highest probabilities with the best independent set. 
 
 
@@ -576,7 +579,7 @@ def f(params):
 #
 # where :math:`k = 1, 2, \dots, p`.
 #
-# In the DQVA, whenever the :math:`j`-th bit of the initial state is one,
+# In the DQVA, whenever the :math:`j`-th bit of the initial state is 1,
 # the corresponding parameter :math:`\alpha_k^j` is set to 0. For example,
 # if the initial state is :math:`\vert 01101 \rangle`, then
 #