# Source code for qiskit.quantum_info.analysis.distance

```
# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2019.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""A collection of discrete probability metrics."""
import numpy as np
[docs]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:
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
[docs]def hellinger_fidelity(dist_p, dist_q):
"""Computes the Hellinger fidelity between
two counts distributions.
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.
dist_q (dict): Second dict of counts.
Returns:
float: Fidelity
Example:
.. jupyter-execute::
from qiskit import QuantumCircuit, execute, BasicAer
from qiskit.quantum_info.analysis import hellinger_fidelity
qc = QuantumCircuit(5, 5)
qc.h(2)
qc.cx(2, 1)
qc.cx(2, 3)
qc.cx(3, 4)
qc.cx(1, 0)
qc.measure(range(5), range(5))
sim = BasicAer.get_backend('qasm_simulator')
res1 = execute(qc, sim).result()
res2 = execute(qc, sim).result()
hellinger_fidelity(res1.get_counts(), res2.get_counts())
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
```