Update the definition of the Hellinger fidelity (bp #5595) (#5703)

* Update the definition of the Hellinger fidelity (#5595)

* Update the definition of the Hellinger fidelity

* Updates based on review

* add explanation

* Update releasenotes/notes/fix_hellinger_fidelity_def-7f60d53e3577fdaf.yaml

Co-authored-by: Christopher J. Wood <cjwood@us.ibm.com>

Co-authored-by: Christopher J. Wood <cjwood@us.ibm.com>
Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>
(cherry picked from commit fc77b19)

* Make hellinger_distance() private and update releasenotes

In #5595 the hellinger_fidelity() method was split into a function to
calculate the hellinger distance and the fidelity function was update to
use that internally. However, we don't want to backport a new feature
in a backport/bugfix release so this commit marks it as private and
updates the release notes to not mention the new function. We will
introduce the new api function in the 0.17.0 release.

Co-authored-by: Paul Nation <nonhermitian@gmail.com>
Co-authored-by: Matthew Treinish <mtreinish@kortar.org>

qiskit/quantum_info/analysis/distance.py

@@ -14,13 +14,55 @@
 import numpy as np
 
 
+def _hellinger_distance(dist_p, dist_q):
+    """Computes the Hellinger distance between
+    two counts distributions.
+
+    Parameters:
+        dist_p (dict): First dict of counts.
+        dist_q (dict): Second dict of counts.
+
+    Returns:
+        float: Distance
+
+    References:
+        `Hellinger Distance @ wikipedia <https://en.wikipedia.org/wiki/Hellinger_distance>`_
+    """
+    p_sum = sum(dist_p.values())
+    q_sum = sum(dist_q.values())
+
+    p_normed = {}
+    for key, val in dist_p.items():
+        p_normed[key] = val/p_sum
+
+    q_normed = {}
+    for key, val in dist_q.items():
+        q_normed[key] = val/q_sum
+
+    total = 0
+    for key, val in p_normed.items():
+        if key in q_normed.keys():
+            total += (np.sqrt(val) - np.sqrt(q_normed[key]))**2
+            del q_normed[key]
+        else:
+            total += val
+    total += sum(q_normed.values())
+
+    dist = np.sqrt(total)/np.sqrt(2)
+
+    return dist
+
+
 def hellinger_fidelity(dist_p, dist_q):
     """Computes the Hellinger fidelity between
     two counts distributions.
 
-    The fidelity is defined as 1-H where H is the
-    Hellinger distance.  This value is bounded
-    in the range [0, 1].
+    The fidelity is defined as :math:`\\left(1-H^{2}\\right)^{2}` where H is the
+    Hellinger distance.  This value is bounded in the range [0, 1].
+
+    This is equivalent to the standard classical fidelity
+    :math:`F(Q,P)=\\left(\\sum_{i}\\sqrt{p_{i}q_{i}}\\right)^{2}` that in turn
+    is equal to the quantum state fidelity for diagonal density matrices.
 
     Parameters:
         dist_p (dict): First dict of counts.
@@ -49,27 +91,10 @@ def hellinger_fidelity(dist_p, dist_q):
             res2 = execute(qc, sim).result()
 
             hellinger_fidelity(res1.get_counts(), res2.get_counts())
-    """
-    p_sum = sum(dist_p.values())
-    q_sum = sum(dist_q.values())
-
-    p_normed = {}
-    for key, val in dist_p.items():
-        p_normed[key] = val/p_sum
-
-    q_normed = {}
-    for key, val in dist_q.items():
-        q_normed[key] = val/q_sum
-
-    total = 0
-    for key, val in p_normed.items():
-        if key in q_normed.keys():
-            total += (np.sqrt(val) - np.sqrt(q_normed[key]))**2
-            del q_normed[key]
-        else:
-            total += val
-    total += sum(q_normed.values())
 
-    dist = np.sqrt(total)/np.sqrt(2)
-
-    return 1-dist
+    References:
+        `Quantum Fidelity @ wikipedia <https://en.wikipedia.org/wiki/Fidelity_of_quantum_states>`_
+        `Hellinger Distance @ wikipedia <https://en.wikipedia.org/wiki/Hellinger_distance>`_
+    """
+    dist = _hellinger_distance(dist_p, dist_q)
+    return (1-dist**2)**2

releasenotes/notes/fix_hellinger_fidelity_def-7f60d53e3577fdaf.yaml

@@ -0,0 +1,6 @@
+---
+fixes:
+  - |
+    The definition of the Hellinger fidelity from has been corrected from the
+    previous defition of :math`1-H(P,Q)` to :math:`[1-H(P,Q)**2]**2` so that it
+    is equal to the quantum state fidelity of P, Q as diagonal density matrices.