In recent years, achieving verifiable quantum advantage on a NISQ device has emerged as an important open problem in quantum information. The sampling-based quantum advantages are not known to have efficient verification methods. This paper investigates the verification of quantum advantage from a cryptographic perspective. We establish a strong connection between the verifiability of quantum advantage and cryptographic and complexity primitives, including efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states ($\mathsf{EFI}$), pseudorandom states ($\mathsf{PRS}$), and variants of minimum circuit size problems ($\mathsf{MCSP}$). Specifically, we prove that a) a sampling-based quantum advantage is either verifiable or can be used to build $\mathsf{EFI}$ and even $\mathsf{PRS}$ and b) polynomial-time algorithms for a variant of $\mathsf{MCSP}$ would imply efficient verification of quantum advantages. Our work shows that the quest for verifiable quantum advantages may lead to applications of quantum cryptography, and the construction of quantum primitives can provide new insights into the verifiability of quantum advantages.

翻译：近年来，在含噪声中等规模量子（NISQ）设备上实现可验证的量子优势已成为量子信息领域的一个重要开放问题。基于采样的量子优势目前尚未被证实存在高效的验证方法。本文从密码学角度研究量子优势的验证问题。我们建立了量子优势可验证性与密码学及复杂性原语之间的紧密联系，包括可高效采样、统计距离远但计算不可区分的（混合）量子态对（$\mathsf{EFI}$）、伪随机量子态（$\mathsf{PRS}$），以及最小电路规模问题（$\mathsf{MCSP}$）的变体。具体而言，我们证明：a）基于采样的量子优势要么是可验证的，要么可用于构造$\mathsf{EFI}$甚至$\mathsf{PRS}$；b）针对$\mathsf{MCSP}$变体的多项式时间算法将意味着量子优势的高效验证。我们的研究表明，对可验证量子优势的探索可能推动量子密码学的应用，而量子原语的构造可为量子优势的可验证性提供新的理论视角。