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hellinger_fidelity(dist_p, dist_q)

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Computes the Hellinger fidelity between two counts distributions.

The fidelity is defined as (1H2)2\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 F(Q,P)=(ipiqi)2F(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.


  • dist_p (dict) – First dict of counts.
  • dist_q (dict) – Second dict of counts.



Return type



from qiskit import QuantumCircuit, execute, BasicAer
from qiskit.quantum_info.analysis import hellinger_fidelity
qc = QuantumCircuit(5, 5)
qc.h(2), 1), 3), 4), 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())


Quantum Fidelity @ wikipedia(opens in a new tab) Hellinger Distance @ wikipedia(opens in a new tab)

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