mindquantum.simulator.fidelity

mindquantum.simulator.fidelity(rho: np.ndarray, sigma: np.ndarray)[source]

Calculate the fidelity of two quantum states.

Definition of quantum state fidelity shown as below.

\[F(\rho, \sigma) = \left( \text{tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right)^2\]

where \(\rho\) and \(\sigma\) are density matrices.

If both \(\rho\) and \(\sigma\) are pure, \(\rho=\left|\psi_\rho\right>\!\left<\psi_\rho\right|\) and \(\sigma=\left|\psi_\sigma\right>\!\left<\psi_\sigma\right|\), then

\[F(\rho, \sigma) = \left| \left< \psi_\rho \middle| \psi_\sigma \right> \right|^2\]

Besides, mixing state vector with density matrix as input is also supported.

Parameters

rho (np.ndarray) – One of the quantum state. Support both state vector and density matrix.
sigma (np.ndarray) – Another quantum state. Support both state vector and density matrix.

Returns

numbers.Number, the fidelity of two quantum states.

Examples

>>> from mindquantum.core.circuit import Circuit
>>> from mindquantum.simulator import Simulator, fidelity
>>> circ = Circuit().h(0).x(1, 0)
>>> sim = Simulator('mqmatrix', 2)
>>> sim.apply_circuit(circ)
>>> rho = sim.get_qs()
>>> sim.reset()
>>> sigma = sim.get_qs()
>>> fidelity(rho, sigma)
0.5000000000000001