$\mathsf{BQP} \subseteq \mathsf{IP}$ 不相对化

中文摘要 AI 辅助

我们构造了一个预言机，相对于该预言机，$\mathsf{BQP} \not\subseteq \mathsf{IP}$，解决了量子复杂性理论中一个长期悬而未决的问题。结合 Aaronson 等人的近期工作，我们的工作还首次给出了 $\mathsf{IP}$ 与 $\mathsf{MIP}$ 之间的预言机分离，回答了一个可追溯至 Fortnow 论文的问题。我们的分离基于 Forrelation 问题，其中给定布尔函数 $f$ 和 $g$，目标是确定 $f$ 是否与 $g$ 的傅里叶谱相关。虽然该任务可由查询高效的量子算法解决，但我们证明它不存在具有多项式通信和多项式查询验证者的经典交互式协议。我们的证明基于 (i) 一个新的结构结果，展示了如何用具有小一阶和二阶导数的凸函数来近似 Avg-Max 电路（众所周知，这类电路在预言机设置中捕捉了交互式证明的能力），以及 (ii) 一项新颖的分析，确立了 Aaronson 和 Ambainis 提出的 Forrelation 分布能够欺骗此类函数。我们的结果表明，任何针对 $\mathsf{BQP}$ 的证明者高效的经典交互式协议必须依赖非相对化技术。这可能部分解释了在量子计算的双重高效、无条件可靠的经典验证方面缺乏进展的原因。

英文摘要

We construct an oracle relative to which $\mathsf{BQP} \not\subseteq \mathsf{IP}$, resolving a long-standing open question in quantum complexity theory. Together with recent work due to Aaronson et al., our work also gives the first oracle separation between $\mathsf{IP}$ and $\mathsf{MIP}$, answering a question dating back to Fortnow's thesis. Our separation is based on the Forrelation problem, where given Boolean functions $f$ and $g$, the goal is to determine if $f$ is correlated with the Fourier spectrum of $g$. While this task is solvable by a query-efficient quantum algorithm, we show that it admits no classical interactive protocol with polynomial communication and a polynomial-query verifier. Our proof is based on (i) a new structural result showing how to approximate Avg-Max circuits (which are well-known to capture the power of interactive proofs in the oracular setting) by convex functions with small first and second derivatives and (ii) a novel analysis establishing that the Forrelation distribution suggested by Aaronson and Ambainis fools such functions. Our results imply that any prover-efficient classical interactive protocol for $\mathsf{BQP}$ must rely on non-relativizing techniques. This might serve as a partial explanation for the lack of progress towards doubly-efficient, unconditionally sound classical verification of quantum computation.

发表机构

• Stanford University（斯坦福大学）
• University of California, Berkeley（加州大学伯克利分校）
• Massachusetts Institute of Technology（麻省理工学院）

机构由 AI 辅助整理，请以论文原文为准。