How hard is it to verify a classical shadow?

2025-10-09

AI Summary

This paper introduces and formalizes the “Classical Shadow Validity” (CSV) problem: given a classical shadow ( S ), determine whether it accurately predicts the statistical properties of a target quantum state with respect to a specified set of observables. Using classical shadow protocols based on local Clifford measurements and their higher-dimensional generalizations, and employing computational complexity analysis, we prove that CSV is QMA-complete in general. Furthermore, when the number of observables grows exponentially, CSV becomes the first natural quantum-complete problem for the second level of the polynomial hierarchy, ( mathrm{PP}^2 ). These results reveal the intrinsic computational hardness of classical shadow verification, establish rigorous theoretical complexity bounds for quantum state characterization, and uncover a deep connection between classical shadows and quantum complexity classes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

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 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]. 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.

Problem

• Verifying classical shadow validity is QMA-complete
• Hardness holds for local Clifford measurement protocols
• Establishes first natural complete problem for quantum hierarchy

Innovation

• Verifying classical shadow validity is QMA-complete
• Local Clifford measurements enable efficient classical shadows
• Hardness extends to high-dimensional shadow protocols