walk op docstrings

qualtran/bloqs/qubitization_walk_operator.ipynb

@@ -38,14 +38,16 @@
     "## `QubitizationWalkOperator`\n",
     "Constructs a Szegedy Quantum Walk operator using LCU oracles SELECT and PREPARE.\n",
     "\n",
+    "For a Hamiltonian $H = \\sum_l w_l H_l$ (s.t. $w_l > 0$ and $H_l$ are unitaries),\n",
     "Constructs a Szegedy quantum walk operator $W = R_{L} . SELECT$, which is a product of\n",
     "two reflections $R_{L} = (2|L><L| - I)$ and $SELECT=\\sum_{l}|l><l|H_{l}$.\n",
     "\n",
     "The action of $W$ partitions the Hilbert space into a direct sum of two-dimensional irreducible\n",
     "vector spaces. For an arbitrary eigenstate $|k>$ of $H$ with eigenvalue $E_k$, $|\\ell>|k>$ and\n",
     "an orthogonal state $\\phi_{k}$ span the irreducible two-dimensional space that $|\\ell>|k>$ is\n",
     "in under the action of $W$. In this space, $W$ implements a Pauli-Y rotation by an angle of\n",
-    "$-2arccos(E_{k} / \\lambda)$ s.t. $W = e^{i arccos(E_k / \\lambda) Y}$.\n",
+    "$-2arccos(E_{k} / \\lambda)$ s.t. $W = e^{i arccos(E_k / \\lambda) Y}$,\n",
+    "where $\\lambda = \\sum_l w_l$.\n",
     "\n",
     "Thus, the walk operator $W$ encodes the spectrum of $H$ as a function of eigenphases of $W$\n",
     "s.t. $spectrum(H) = \\lambda cos(arg(spectrum(W)))$ where $arg(e^{i\\phi}) = \\phi$.\n",

qualtran/bloqs/qubitization_walk_operator.py

@@ -53,7 +53,6 @@ class QubitizationWalkOperator(GateWithRegisters):
         select: The SELECT lcu gate implementing $SELECT=\sum_{l}|l><l|H_{l}$.
         prepare: Then PREPARE lcu gate implementing
             $PREPARE|00...00> = \sum_{l=0}^{L - 1}\sqrt{\frac{w_{l}}{\lambda}} |l> = |\ell>$
-        sum_of_lcu_coefficients: value of $\lambda$, i.e. sum of the coefficients $w_l$.
         control_val: If 0/1, a controlled version of the walk operator is constructed. Defaults to
             None, in which case the resulting walk operator is not controlled.
         power: Constructs $W^{power}$ by repeatedly decomposing into `power` copies of $W$.
@@ -97,6 +96,7 @@ def reflect(self) -> ReflectionUsingPrepare:
 
     @cached_property
     def sum_of_lcu_coefficients(self) -> Optional[float]:
+        r"""value of $\lambda$, i.e. sum of the coefficients $w_l$."""
         return self.prepare.l1_norm_of_coeffs
 
     def decompose_from_registers(