Complexity class containments involving interactive proof classes are famously nonrelativizing: although $\mathsf{IP} = \mathsf{PSPACE}$, Fortnow and Sipser showed that that there exists an oracle relative to which $\mathsf{coNP} \not\subseteq \mathsf{IP}$. In contrast, the question of whether the containment $\mathsf{BQP} \subseteq \mathsf{IP}$ is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment $\mathsf{BQP} \subseteq \mathsf{MIP}$ holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle $O$, a $\mathsf{PCP}$ proof system for $\mathsf{BQP}^{O}$ where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle $O$. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an $\mathsf{IP}$ for $\mathsf{BQP}$ in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.

翻译：涉及交互式证明类的复杂性类包含关系以非相对化特性著称：尽管$\mathsf{IP} = \mathsf{PSPACE}$，Fortnow与Sipser证明存在一个谕示使得$\mathsf{coNP} \not\subseteq \mathsf{IP}$。相比之下，$\mathsf{BQP} \subseteq \mathsf{IP}$这一包含关系是否具有相对化特性仍是未解难题。本文通过证明$\mathsf{BQP} \subseteq \mathsf{MIP}$对任意经典谕示成立，在解决该问题上取得进展。我们通过为任意经典谕示$O$构建一个$\mathsf{PCP}$证明系统来获得此结果，其中验证者需对指数长度证明和谕示$O$进行多项式次经典查询。本构造受Grover与Rudolph的状态合成算法启发，补充了Aharonov、Arad和Vidick提出的"指数级PCP"——该成果基于不同思路且不具备相对化特性。我们提议将相对化作为证明者效率的代理指标，并希望面向谕示世界中$\mathsf{BQP}$的$\mathsf{IP}$研究进展，能引导出非相对化世界中用于向经典怀疑者证明任意量子计算的非密码学交互协议，这亦是量子复杂性理论中一个长期未决的开放问题。