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.

翻译：经典阴影是量子态的简洁经典表示，它允许编码量子态ρ的一组性质P，同时仅需对ρ进行与P规模对数相关的测量副本。在本工作中，我们从计算复杂性的角度首次研究了经典阴影的验证问题（称为经典阴影有效性，CSV），其核心问题是：给定一个经典阴影S，验证S能否预测量子态的测量统计量有多困难？我们首先证明，即使对于[Huang, Kueng, Preskill, Nature Physics 2020]提出的利用局部Clifford测量的优雅简洁经典阴影协议，CSV也是QMA完全的。这种困难性对于[Mao, Yi, and Zhu, PRL 2025]提出的上述协议的高维扩展同样成立。相比之下，我们证明对于利用全局Clifford测量的HKP和MYZ协议，CSV对于低秩可观测量可以被“去量子化”，即在标准采样假设下通过随机多项式时间算法求解。在其他结果中，我们还证明对于指数多个可观测量，CSV是多项式层次第二层级量子推广的完全问题，这为该复杂性类提供了首个自然完全问题。