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according to the paper https://doi.org/10.1038/s42254-022-00535-2 , the advantage of classical shadows is doing measurements first and asking questions later.

But in real experiments, who would do measurement first and ask questions later, being quantum resources so expensive?

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If you know what questions to ask beforehand, you can often do much better than with classical shadows / randomized measurements, at least for Pauli observables. You can, e.g., run a derandomization approach (cf. https://arxiv.org/abs/2103.07510) and get away with many fewer samples. For many-body Hamiltonians there are also grouping strategies for the Pauli terms that typically perform better. Nevertheless, there is also a combination of classical shadows with grouping that combines the advantages of the two approaches (https://arxiv.org/abs/2301.03385). For a single or a few Pauli observables, direct estimation is also better. Thus, in these cases, one should not use classical shadows blindly, as it can lead to a measurement overhead of a few orders of magnitude.

Nevertheless, there are more than a few reasons why classical shadows make sense:

  1. You can easily make changes to the observables and re-run the estimation with the same data.

  2. Entanglement detection (http://arxiv.org/abs/2007.06305, https://arxiv.org/abs/2311.08108). I am not sure whether there are better performing alternatives.

  3. As a tool for noise estimation. You can use classical shadows to estimate matrix entries of quantum channels and use this to, e.g., learn Hamiltonians or Lindbladians (http://arxiv.org/abs/2205.09567). In this case you only have limited information about the quantities to be estimated, so the randomized shadow scheme is likely optimal.

  4. If you want to estimate non-local observables (e.g., fidelities), it is not entirely clear whether you can do much better. For fidelities, an alternative would be direct fidelity estimation, which, however, may have worse sampling complexity then fidelity estimation via classical shadows (https://arxiv.org/abs/2204.02995).

  5. As a theoretical tool for a theory of quantum learning (see, e.g., https://www.science.org/doi/10.1126/science.abn7293).

  6. The shadow framework is interesting and has plenty of connections to other estimation protocols. For a glimpse, see https://www.nature.com/articles/s41467-023-39382-9.

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    $\begingroup$ I'd also add to the last point, regarding understanding what classical shadows are all about from a more general estimation perspective, arxiv.org/abs/2301.13229, arxiv.org/abs/2205.08990, and arxiv.org/abs/2105.05992 $\endgroup$
    – glS
    Mar 5 at 11:09
  • $\begingroup$ @glS To clarify, my understanding of shadows is via general frame theory. The underlying mechanism has already been laid out in arxiv.org/abs/quant-ph/0604049 . What I meant in 6 was that the post-processing can be changed to perform other estimation tasks, beyond linear/polynomial functions in the state. $\endgroup$ Mar 6 at 9:17
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The idea is that with a suitable choice of measurement scheme, you can ask essentially any question "later", and estimate the information you're looking for efficiently. In other words, you can fix the measurement strategy, if that alone allows to recover any observable of interest about input states. Classical shadows are just estimators defined on individual measurement outcomes. If the measurement is informationally complete, as these schemes usually are, then any piece information is recoverable, with sufficient statistics.

You can prove that for suitable choices of measurement settings, you can estimate arbitrary observables with a number of resources that does not depend on the dimension of the measured state, which is what you'd consider "efficient" in this context. You can in fact estimate many observables at the same time in this way. You can estimate $M$ observables with a probability $\delta$ of the max error being less than $\epsilon$ with something like (and I'm going by memory here) $N\sim \log(M) \log(2/\delta)/\epsilon^2$. This is using measurement schemes that are very symmetric, such as the original "random projective measurements" scheme of Huang et al. That's efficient enough that it can make sense to not tailor the measurement apparatus to recover a specific observable, unless you really need to optimise for that.

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