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On Classical and Hybrid Shadows of Quantum States

by Saumya Shivam, C. W. von Keyserlingk, S. L. Sondhi

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Submission summary

Authors (as registered SciPost users): Saumya Shivam
Submission information
Preprint Link:  (pdf)
Date submitted: 2022-06-15 02:34
Submitted by: Shivam, Saumya
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


Classical shadows are a computationally efficient approach to storing quantum states on a classical computer for the purposes of estimating expectation values of local observables, obtained by performing repeated random measurements. In this note we offer some comments on this approach. We note that the resources needed to form classical shadows with bounded relative error depend strongly on the target state. We then comment on the advantages and limitations of using classical shadows to simulate many-body dynamics. In addition, we introduce the notion of a hybrid shadow, constructed from measurements on a part of the system instead of the entirety, which provides a framework to gain more insight into the nature of shadow states as one reduces the size of the subsystem measured, and a potential alternative to compressing quantum states.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 1 on 2022-9-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2206.06616v1, delivered 2022-09-03, doi: 10.21468/SciPost.Report.5633


1) Paper is a generally accessible treatment of quantum tomography for condensed matter theorists.

2) Provides some technical results on the application of shadow tomography to classical compression of quantum states.


1) The paper does not provide convincing evidence that hybrid shadows are useful, nor does it place the idea in the broader context of low-rank approximations of quantum states by sums of stabilizer states.

2) The paper does not give a comprehensive treatment of the subject and fails to discuss a nontrivial version of classical shadows.


The paper under consideration provides some commentary on classical shadows. The main contribution is the introduction of hybrid shadows that are relevant for simulation of quantum systems on classical computers. However, it is worth noting that hybrid shadows do not seem to present any asymptotic improvements in the classical cost of simulation.

I have a several suggestions and comments for how to improve the discussion and results in the paper, making it more closely align with the state-of-the-art in quantum information.

1) The idea of using local measurements to characterize quantum states is the basis for quantum state tomography, which is textbook material in quantum information. The primary technical advance of Ref. [2] was to show that one could apply similar techniques to operators with bounded Hilbert-Schmidt norm. However, this method is only possible and computationally efficient when one uses a global random Clifford to form the shadows. The present paper does not discuss or analyze this global shadows approach and instead focuses only on local shadows without justification. This oversight leads to some erroneous or misleading conclusions in the paper as global shadows would allow the sample efficient representation of time-evolved operators. The Hilbert-Schmidt norm is invariant under unitary evolution.

2) In section IIIA, the authors discuss the disadvantage of using local shadows to estimate commuting Paulis over direct measurement of commuting operators. This limitation is well known in quantum tomography and can be overcome by instead performing Bell basis measurements on multiple-copies of the state rho. The extension of the shadow formalism to this case was provided in Huang, Kueng and Preskill PRL (2021), where it was shown how to obtain quantum advantage over single-copy classical shadows using two-copy measurements. It is a very simple, but powerful idea. The operators P x P are commuting for any Pauli P, therefore one can measure [trace[P rho]]^2 efficiently using Bell basis measurements.

3) Relating to the discussion of hybrid shadows, the idea of using stabilizer states to compress quantum states is an old idea in quantum information, which can be traced back to the theory of magic in fault-tolerance. As two representative references, I provide Bravyi and Gosset PRL (2016) and Gosset and Smolin, arXiv:1801.05721 (2018). The hybrid shadows discussed in the present paper do not seem to have any asymptotic advantages over just using classical shadows or other standard compression techniques. Perhaps hybrid shadows could have some advantage in non-asymptotic settings, but the authors do not provide any strong evidence that this is the case.

I would recommend that the paper be modified to address these points. With such revisions it could be suitable for publication. As it stands, I think the paper is lacking sufficient context or results to be useful to the condensed matter theory community (which is its stated goal).

Requested changes

1 - provide a systematic study and analysis of global shadows.

2 - provide appropriate commentary on the role of entangling measurements such as simple Bell basis measurements in overcoming the limitations of classical shadows. Perhaps several other results in the paper also need to be revisited in light of this perspective.

3 - provide convincing evidence that hybrid shadows present some asymptotic or non-asymptotic advantage in classical simulation of quantum systems. Compare to methods based on representations of quantum states by low rank stabilizer approximations.

  • validity: good
  • significance: ok
  • originality: ok
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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