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Consider a pure $n$-qubit quantum state $|\psi_\theta\rangle$ prepared by some parametrized quantum circuit. There exist well-known algorithms to efficiently estimate the quantum Fisher information matrix $F(\theta)$. Is any algorithm known for efficiently estimating the classical Fisher information associated with the measurement outcomes relative to the standard basis; that is, for the probability distribution $p_\theta(x) := |\langle x | \psi_\theta \rangle|^2$?

By efficient, I mean scaling polynomially with respect to the number of qubits $n$. In the ideal case, it should be as efficient as estimating the QFI.

Edit: An extremely naive approach is just to do density estimation on the outcome probabilities. But this is just a heuristic whose scaling is difficult to characterize.

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  • $\begingroup$ Edits done @glS $\endgroup$
    – phonon
    Sep 18, 2023 at 1:25
  • $\begingroup$ thanks, that helps. Are you asking about a quantum algorithm to compute the (classical) Fisher info? $\endgroup$
    – glS
    Sep 18, 2023 at 1:26
  • $\begingroup$ Well, I’ll be extremely impressed if you manage to classically calculate it efficiently :) So yes, we are allowed to assume quantum resources. [merely sampling from n-qubit PQCs is classically hard] $\endgroup$
    – phonon
    Sep 18, 2023 at 1:29
  • $\begingroup$ I hate to ask the same question as @glS but: can we have a quantum circuit find $p_\theta(x)$ for all $x$ and then just take a function of those classical data on our classical computer? $\endgroup$ Sep 18, 2023 at 18:49
  • $\begingroup$ Is the point that there may be a more efficient calculation that directly finds the quantities of interest without measuring each of the probabilities first? $\endgroup$ Sep 18, 2023 at 18:50

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Section 3 of this paper talks about strategies to obtain the classical Fisher information in a NISQ computing setting.

More concretely, this recent preprint details how to measure it in practice with resources quadratically smaller than that required for the QFI. The idea is that for the CFI you "just" need to estimate the derivatives of the probability distribution (of which there are linearly many, in terms of parameter count) and the probabilities itself. For the QFI one has to run quadratically many additional circuits (again, in terms of the parameter count). As far as I am aware, convergence can be quite bad if the probability distribution has a lot of entries close to its largest entries.

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