I'm trying to estimate the fidelity of some family of unitaries $U(\theta)$ implemented on a noisy quantum computer. To do so, I start from an uninitialized state $|0\rangle$ and run the circuit $U(\theta)^\dagger U(\theta)$, and then measure the probability of returning to the all-zeros bitstring as
$$ P_\theta (0) = |\langle 0| \mathcal{E}_{U(\theta)^\dagger} \mathcal{E}_{U(\theta)} |0 \rangle |^2 $$
where $\mathcal{E}_U$ is the (noisy) process that results from trying to run $U$. If $P(0) \neq 1$ then I have some idea that my operation ran with lower fidelity. But I would like to know more about how well this kind of reasoning generalizes:
- How well does $P_\theta(0)$ track the fidelity of this process, $F(\mathcal{E}_{U(\theta)} , U(\theta))$, if at all?
- Can I expect any kind of relationship between $P_\theta(0)$ and $P_\theta(\psi) \equiv |\langle \psi| \mathcal{E}_{U(\theta)^\dagger} \mathcal{E}_{U(\theta)} |\psi \rangle |^2$, i.e. the same experiment repeated for a different input state?
- Can I expect any kind of relationship between $P_\theta(0)$ and $P_\phi (0)\equiv |\langle 0| \mathcal{E}_{U(\phi)^\dagger} \mathcal{E}_{U(\phi)} |0 \rangle |^2$ i.e. the same experiment repeated on a similar unitary $U(\phi)$
I can think of specific counterexamples that show that some of these relationships fail to hold but ideally there would be a way to assess the likelihood of this being a good (or bad!) metric.