发表机构
National Taiwan University; National Tsing Hua University. National Center for Excellence in Quantum Information Science and Engineering.（台湾大学; 国立清华大学。量子资讯科学与工程卓越中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明量子建议下多项式时间量子计算无法保持状态规模进行错误缩减，需更长建议，并给出具体分离结果。
AI 中文摘要
Marriott和Watrous证明了量子Merlin-Arthur博弈在无需增加见证规模的情况下可实现通用错误缩减[计算复杂性，2005]。在本工作中，我们表明这种保持状态规模的放大性质对于带量子建议的多项式时间量子计算并不成立。特别地，我们提出了若干判定问题，对于这些问题，即使是消失的加性错误缩减也需要更长的建议。更精确地，对于每个多项式有界的建议长度$m(n)\geq n^4$和每个误差界$\varepsilon(n)$（其低于$1/2$至少一个逆多项式量），存在一个正函数$\delta$，满足$\delta(n)=O\bigl(\min\{(\log m/m)^{1/4},\\ \sqrt{\log m/m}\\,/(1/2-\varepsilon(n))\}\bigr)$，使得$\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + \delta}/\mathsf{q}m$；对于常数$\varepsilon$，差距为$O(\sqrt{\log m/m})$。这里，$\mathsf{BQP}_\varepsilon/\mathsf{q}m$是语言类，其由多项式时间量子算法识别，错误至多为$\varepsilon(n)$，算法使用一个仅依赖于输入长度$n$的$m(n)$量子比特建议态。我们通过证明更强的分离$\mathsf{P}_{\varepsilon+\delta}/\mathsf{r}m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q}m$来展示这一点，其中$\mathsf{P}_{\varepsilon}/\mathsf{r}m$是语言类，其由确定性多项式时间算法识别，错误至多为$\varepsilon(n)$，算法使用一个从仅依赖于$n$的分布中采样的$m(n)$比特建议串。
英文摘要
Marriott and Watrous showed that quantum Merlin--Arthur games admit generic error reduction without increasing witness size [Computational Complexity, 2005]. In this work, we show that this state-size-preserving amplification property does not hold for polynomial-time quantum computation with quantum advice. In particular, we present decision problems for which even a vanishing additive error reduction requires longer advice. More precisely, for every polynomially bounded advice length $m(n)\geq n^4$ and every error bound $\varepsilon(n)$ that stays below $1/2$ by at least an inverse polynomial, there is a positive function $δ$ with $δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)$ such that $\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}m$; for constant $\varepsilon$ the gap is $O(\sqrt{\log m/m})$. Here, $\mathsf{BQP}_\varepsilon/\mathsf{q}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a polynomial-time quantum algorithm with an $m(n)$-qubit advice state that only depends on the input length $n$. We show this by proving a stronger separation $\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m$, where $\mathsf{P}_{\varepsilon}/\mathsf{r}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a deterministic polynomial-time algorithm with an $m(n)$-bit advice string sampled from a distribution that depends only on $n$.
