发表机构
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究两消息量子交互式证明中简洁证明者的能力，引入类QIP_{ℓ-bit}(2)，通过量子态可区分性给出完全刻画，证明其某些情形坍缩为QSZK，并显示量子公共随机币使交互无用，解决Sahai-Vadhan开放问题。
AI 中文摘要
Goldreich、Vadhan和Wigderson（CC，2002）研究的具有简洁证明者的交互式证明系统，在经典设置中刻画了可通过对数级证明者通信验证的问题。对于两消息量子类比，即使单比特证明者响应也包含量子统计零知识（$\sf QSZK$），该概念由Watrous（FOCS 2002）引入。然而，将验证者的提问限制为经典公共随机币会使相应类坍缩为$\sf BQP$，如Beigi、Shor和Watrous（ToC，2011）所示。我们进一步研究具有简洁证明者的两消息量子交互式证明系统。为此，我们引入类${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$，其中$\ell$是证明者响应的长度，并建立：1. 通过多态可区分性对${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$的自然完全刻画。特别地，量子态可区分性（QSD）是${\sf QIP}_{\rm bit}$完全的。由于QSD是$\sf QSZK$困难的，我们的结果将${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$（对于$\ell\geq 2$）置于“略高于”$\sf QSZK$的图景中。2. ${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$坍缩为$\sf QSZK$的简单情形。我们证明当$a(n)-b(n)\geq 1/O(\log n)$时，QSD$[a,b]$（从而${\sf QIP}_{\rm bit}[a,b]$）属于$\sf QSZK$，并将其与从${\sf QIP}_{\ell\text{-}{\rm bit}}[2,c,s]$到${\sf QIP}_{\rm bit}$的答案压缩相结合，获得另一个简单情形：当$2c>(1+2^{\ell/2})s$时。值得注意的是，我们改进的极化适用于SD和$\sf SZK$，解决了Sahai和Vadhan（JACM，2003）中的一个开放问题。3. 量子公共随机币也使交互无用：具有常数间隙的${\sf qc}\text{-}{\sf QAM}[O(\sqrt{\log{n}})]$属于$\sf BQP$，其中${\sf qc}\text{-}{\sf QAM}[\ell]$是${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$的一个子类，其中验证者的提问恰好是EPR对的一半。
英文摘要
Interactive proof systems with a laconic prover, studied by Goldreich, Vadhan, and Wigderson (CC, 2002), capture problems verifiable with logarithmic prover communication in the classical setting. For two-message quantum analogs, even a single-bit prover response contains quantum statistical zero-knowledge ($\sf QSZK$), introduced by Watrous (FOCS 2002). However, restricting the verifier's question to classical public coins collapses the corresponding class to $\sf BQP$, as shown by Beigi, Shor, and Watrous (ToC, 2011). We further study two-message quantum interactive proof systems with a laconic prover. To this end, we introduce the class ${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$, where $\ell$ is the length of the prover's response, and establish: 1. A natural complete characterization of ${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$ by Multi-State Distinguishability. In particular, Quantum State Distinguishability (QSD) is ${\sf QIP}_{\rm bit}$-complete. Since QSD is $\sf QSZK$-hard, our result places ${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$, for $\ell\geq 2$, in a landscape "just above" $\sf QSZK$. 2. Easy regimes for ${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$ collapsing to $\sf QSZK$. We prove that QSD$[a,b]$ (and thus ${\sf QIP}_{\rm bit}[a,b]$) is in $\sf QSZK$ when $a(n)-b(n)\geq 1/O(\log n)$, and combine this with an answer compression from ${\sf QIP}_{\ell\text{-}{\rm bit}}[2,c,s]$ to ${\sf QIP}_{\rm bit}$ to obtain another easy regime when $2c>(1+2^{\ell/2})s$. Remarkably, our improved polarization applies to SD and $\sf SZK$, resolving an open problem in Sahai and Vadhan (JACM, 2003). 3. Quantum public coins also make the interaction useless: ${\sf qc}\text{-}{\sf QAM}[O(\sqrt{\log{n}})]$ with constant gap is in $\sf BQP$, where ${\sf qc}\text{-}{\sf QAM}[\ell]$ is a subclass of ${\sf QIP}_{\ell\text{-}{\rm bit}}(2)$ in which the verifier's question is exactly halves of EPR pairs.
Comments66 pages, 4 protocols, 2 algorithms, 3 circuits, 4 tables. v2: Minor changes
