Fidelity calculation for density matrices improvement

https://github.com/quantumlib/Cirq/blob/782c14e0cc821794d21e82f30df8e6998f4b9992/cirq-core/cirq/qis/measures.py#L239-L242

The current implementation uses the direct definition of fidelity, which works well for small dimensions. From my experience, I was getting values significantly greater than 1. For the fidelity between rho1 and rho2, I found that squaring the sum of the singular values of the product of square_root(rho1) and square_root(rho2) gives a more accurate calculation. This equation is equal to fidelity. The idea is not mine and came from qutip.

This is my version of the implementation that I use.

import scipy
def get_fidelity(rho1, rho2):
    rho1_sqrt=scipy.linalg.sqrtm(rho1)
    rho2_sqrt=scipy.linalg.sqrtm(rho2)
    return scipy.linalg.svdvals(rho1_sqrt @ rho2_sqrt).sum()**2

[tanujkhattar]

@anonymousr007 Yes, the issue is still open.

The goal is to replace the current implementation of calculating fidelity of two density matrices with the improved implementation that has lower numerical error.

Current implementation: https://github.com/quantumlib/Cirq/blob/782c14e0cc821794d21e82f30df8e6998f4b9992/cirq-core/cirq/qis/measures.py#L239-L242

New implementation:

def get_fidelity(rho1, rho2):
    rho1_sqrt=scipy.linalg.sqrtm(rho1)
    rho2_sqrt=scipy.linalg.sqrtm(rho2)
    return scipy.linalg.svdvals(rho1_sqrt @ rho2_sqrt).sum()**2

The PR implementing the change should also add a unit test demonstrating improved numerical stability of the new implementation, s.t. the current implementation fails but the new one passes.

[daxfohl]

Nice, I noticed pretty bad rounding errors when testing https://github.com/quantumlib/Cirq/pull/5265 and wasn’t sure what to make of it. (I had to reduce the atol in my tests to just 1e-2)