How hard is it to verify a classical shadow?
Georgios Karaiskos, Dorian Rudolph, Johannes Jakob Meyer, Jens Eisert, Sevag Gharibian
TL;DR
The paper investigates the computational complexity of verifying classical shadows (CSV) across multiple shadow protocols, establishing QMA-hardness for HKP and MYZ in polynomial-observable settings and a poly-time randomizable dequantization for global Clifford measurements under Frobenius-norm bounds. It further shows that CSV with exponentially many observables is qc-Σ2-complete, connecting CSV to the quantum-classical polynomial hierarchy, and introduces product-state variants that align with QMA(2) and qc-Σ2(2). To bridge theory and practice, the authors leverage matrix-sketching and SDP techniques to realize dequantization results and provide robustness and multi-shadow formulations (SampleCSV, MCSV/BLOC) with corresponding reductions. Overall, the work maps the intricate landscape of CSV hardness and tractable regimes, highlighting limitations of shadow-based verification and outlining a roadmap for future exploration of additional shadow protocols and higher observables.
Abstract
Classical shadows are succinct classical representations of quantum states which allow one to encode a set of properties P of a quantum state rho, while only requiring measurements on logarithmically many copies of rho 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-rank observables, i.e., solved in randomized poly-time with standard sampling assumptions. Among other results, we also 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.