We generalize quantum-classical PCPs, first introduced by Weggemans, Folkertsma and Cade (TQC 2024), to allow for $q$ quantum queries to a polynomially-sized classical proof ($\mathsf{QCPCP}_{Q,c,s}[q]$). Exploiting a connection with the polynomial method, we prove that for any constant $q$, promise gap $c-s = \Omega(1/\text{poly}(n))$ and $\delta>0$, we have $\mathsf{QCPCP}_{Q,c,s}[q] \subseteq \mathsf{QCPCP}_{1-\delta,1/2+\delta}[3] \subseteq \mathsf{BQ} \cdot \mathsf{NP}$, where $\mathsf{BQ} \cdot \mathsf{NP}$ is the class of promise problems with quantum reductions to an $\mathsf{NP}$-complete problem. Surprisingly, this shows that we can amplify the promise gap from inverse polynomial to constant for constant query quantum-classical PCPs, and that any quantum-classical PCP making any constant number of quantum queries can be simulated by one that makes only three classical queries. Nevertheless, even though we can achieve promise gap amplification, our result also gives strong evidence that there exists no constant query quantum-classical PCP for $\mathsf{QCMA}$, as it is unlikely that $\mathsf{QCMA} \subseteq \mathsf{BQ} \cdot \mathsf{NP}$, which we support by giving oracular evidence. In the (poly-)logarithmic query regime, we show for any positive integer $c$, there exists an oracle relative to which $\mathsf{QCPCP}[\mathcal{O}(\log^c n)] \subsetneq \mathsf{QCPCP}_Q[\mathcal{O}(\log^c n)]$, contrasting the constant query case where the equivalence of both query models holds relative to any oracle. Finally, we connect our results to more general quantum-classical interactive proof systems.

翻译：我们将由 Weggemans、Folkertsma 和 Cade（TQC 2024）首次提出的量子-经典 PCP 推广到允许对多项式规模经典证明进行 $q$ 次量子查询（$\mathsf{QCPCP}_{Q,c,s}[q]$）。利用与多项式方法的联系，我们证明对于任意常数 $q$、承诺间隙 $c-s = \Omega(1/\text{poly}(n))$ 和 $\delta>0$，有 $\mathsf{QCPCP}_{Q,c,s}[q] \subseteq \mathsf{QCPCP}_{1-\delta,1/2+\delta}[3] \subseteq \mathsf{BQ} \cdot \mathsf{NP}$，其中 $\mathsf{BQ} \cdot \mathsf{NP}$ 是通过量子归约到 $\mathsf{NP}$ 完全问题的承诺问题类。令人惊讶的是，这表明对于常数查询的量子-经典 PCP，我们可以将承诺间隙从逆多项式放大到常数，并且任何进行常数次量子查询的量子-经典 PCP 都可以通过仅进行三次经典查询的 PCP 来模拟。然而，尽管我们能够实现承诺间隙放大，我们的结果也给出了强有力的证据表明不存在针对 $\mathsf{QCMA}$ 的常数查询量子-经典 PCP，因为 $\mathsf{QCMA} \subseteq \mathsf{BQ} \cdot \mathsf{NP}$ 的可能性很低，我们通过提供启示性证据来支持这一观点。在（多）对数查询机制中，我们证明对于任意正整数 $c$，存在一个启示器使得 $\mathsf{QCPCP}[\mathcal{O}(\log^c n)] \subsetneq \mathsf{QCPCP}_Q[\mathcal{O}(\log^c n)]$，这与常数查询情形形成对比——在常数查询情形下，两种查询模型相对于任意启示器都是等价的。最后，我们将结果与更一般的量子-经典交互式证明系统联系起来。