2026-09-08T02:22:06Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26512 2026-09-05T19:46:10Z ddc:004 open_access

How Hard Is It to Verify a Classical Shadow? Karaiskos, Georgios Rudolph, Dorian Meyer, Johannes Jakob Eisert, Jens Gharibian, Sevag classical shadows quantum complexity theory QMA quantum polynomial hierarchy 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Georgios Karaiskos and Dorian Rudolph and Johannes Jakob Meyer and Jens Eisert and Sevag Gharibian 2026 Is Part Of LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2026.123 urn:nbn:de:0030-drops-265121 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.123 eng https://creativecommons.org/licenses/by/4.0/legalcode