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How Hard Is It to Verify a Classical Shadow?

Authors Georgios Karaiskos , Dorian Rudolph , Johannes Jakob Meyer , Jens Eisert , Sevag Gharibian

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Quantum complexity theory
Keywords
  • classical shadows
  • quantum complexity theory
  • QMA
  • quantum polynomial hierarchy

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Abstract

Classical shadows are succinct classical representations of quantum states which allow one to encode a set of properties P of a quantum state ρ, while only requiring measurements on logarithmically many copies of ρ in the size of P. In this work, we initiate the study of verification of classical shadows, denoted classical shadow validity (CSV), from the perspective of computational complexity, which asks: Given a classical shadow S, how hard is it to verify that S predicts the measurement statistics of a quantum state? We first show that even for the elegantly simple classical shadow protocol of [Huang, Kueng, Preskill, Nature Physics 2020] utilizing local Clifford measurements, CSV is QMA-complete. This hardness continues to hold for the high-dimensional extension of said protocol due to [Mao, Yi, and Zhu, PRL 2025]. In contrast, we show that for the HKP and MYZ protocols utilizing global Clifford measurements, CSV can be "dequantized" for low-Frobenius norm observables, i.e., solved in randomized poly-time with standard sampling assumptions. Finally, we show that CSV for exponentially many observables is complete for a quantum generalization of the second level of the polynomial hierarchy, yielding the first natural complete problem for such a class.

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Georgios Karaiskos, Dorian Rudolph, Johannes Jakob Meyer, Jens Eisert, and Sevag Gharibian. How Hard Is It to Verify a Classical Shadow?. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 123:1-123:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.123

Author Details

Georgios Karaiskos
  • Department of Computer Science, Paderborn University, Germany
Dorian Rudolph
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany
Johannes Jakob Meyer
  • Freie Universität Berlin, Germany
Jens Eisert
  • Freie Universität Berlin, Germany
Sevag Gharibian
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany

Funding

  • Eisert, Jens: Supported by the BMFTR (QSolid, Hybrid++, QuSol, MUNIQC-Atoms, QuSol, PasQuops, Hybrid++), the Munich Quantum Valley (K-4 and K-8), the Quantum Flagship (PasQuans2, Millenion), QuantERA (HQCC), the Clusters of Excellence MATH+ and ML4Q, the DFG (CRC 183, SPP 2514), Berlin Quantum, and the ERC (DebuQC).
  • Gharibian, Sevag: Supported by the DFG under grant numbers 563388236 and 450041824, the BMFTR via project PhoQuant (grant number 13N16103), and the project PhoQC from the programme Profilbildung 2020 of the Ministry of Culture and Science of the State of North Rhine-Westphalia.

Acknowledgements

We thank Asad Raza for helpful discussions.

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