Add `StabilizerState` support to `state_fidelity`

[chriseclectic]

What is the expected enhancement?

Currently the StabilizerState class is not supported by the state_fidelity function which can only handle Statevector and DensityMatrix states.

This function should be extended to have an efficient form of computing the state fidelity between two stabilizer states, and a (probably inefficient) version for computing the fidelity between a stabilizer state and a non-stabilizer state (statevector or density matrix).

Possible implementation

I'm not sure if there is a more efficient method but one way you could compute the fidelity between two stabilizer states |a> = A|0>, |b> = B|> for cliffords A, B would be to compute the the all-zero measurement outcome probability for the stabilizer state B^\dagger A|0>as p = |<0|B^\dagger A|0>|^2 = F(|a>, |b>) .

For the second case one way would be allowing conversion of a StabilizerState to Statevector. This could be implemented (inefficiently) by running the simulation for the Clifford circuit on the Statevector class. This would also generally useful functionality to add to the StabilzerState class.

[ikkoham]

The efficient method to calculate an inner product between two stabilizer states is described in the last of Sec. Ⅲ https://arxiv.org/abs/quant-ph/0406196.

[ShellyGarion]

Does this depend on #7245 and #7269 being merged?

[patdenim]

Can this function be extended to compute the state fidelity between twosuperposition of stabilizer states too? not only between 2 Stabilizer states..... This is a request from Keio university.

[anonymousr007]

Hello, I am Rishabh, an undergraduate final year computer engineering student. This issue is still open or not? My Interests are:-

• Quantum Information Science.
• Quantum Error Corrections.