发表机构
Portland State University（波特兰州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过分数块灵敏度统一推导了多种带噪预言机模型下的量子查询复杂度下界，并发现疏忽噪声不破坏量子加速，同时给出混合查询与退相噪声下的新界。
AI 中文摘要
我们研究了在几种不完美预言机访问模型下的量子查询复杂度，并通过一个基于分数块灵敏度（fractional block sensitivity, \fbs）的统一框架推导出下界。对于以概率 \\(1-p\\) 应用正确查询的疏忽型预言机，我们证明了以 \fbs(f,0^n) 表示的下界。我们还表明，或许令人惊讶的是，疏忽并不一定会破坏量子加速：任何函数 \\(f\\) 都可以转化为一个部分函数 \\(f'\\)，其疏忽查询复杂度本质上保留了 \\(f\\) 的量子查询复杂度。这给出了即使在疏忽查询下也具有指数级量子加速的部分函数，并且特别排除了在该模型中对部分函数以 \fbs(f) 表示的通用下界。对于另外两种模型，我们获得了对所有布尔函数以 \fbs(f) 表示的通用下界。对于使用 \\(Q\\) 次相干查询和 \\(C\\) 次经典查询的混合算法，我们证明了权衡关系 \\(C+Q^2=\Omega(\fbs(f))\\)。对于一种独立同分布（IID）退相噪声模型，其中每次查询独立地以速率 \\(p\\) 使查询索引寄存器退相，我们证明了需要 \\(\Omega\\!\left(p\\,\fbs(f)\right)\\) 次查询。最后，我们引入了更广泛的一类时变退相模型，并确定了一个始终以 \fbs(f) 为下界的变分资源。计算该资源可归结为一个凸优化问题，为推导新噪声调度下的下界提供了一种简单方法。作为应用，我们恢复了混合和IID退相界，并确定了当退相速率随时间增长时非结构化搜索的查询复杂度。
英文摘要
We study quantum query complexity under several models of imperfect oracle access, and develop lower bounds through a common framework based on fractional block sensitivity ($\operatorname{fbs}$). For a negligent oracle that applies the correct query with probability $1-p$, we prove a lower bound in terms of $\operatorname{fbs}(f,0^n)$. We also show, perhaps surprisingly, that negligence need not destroy quantum speedups: any function $f$ can be transformed into a partial function $f'$ whose negligent query complexity essentially preserves the quantum query complexity of $f$. This gives partial functions with exponential quantum speedups even under negligent queries, and in particular rules out a general lower bound in terms of $\operatorname{fbs}(f)$ for partial functions in this model. For two other models, we obtain general lower bounds in terms of $\operatorname{fbs}(f)$ for all Boolean functions. For hybrid algorithms using $Q$ coherent and $C$ classical queries, we prove the tradeoff $C+Q^2=Ω(\operatorname{fbs}(f))$. For an IID dephasing noisy model where each query dephases the query-index register at rate $p$ independently, we prove $Ω\!\left(p\,\operatorname{fbs}(f)\right)$ queries are necessary. Finally, we introduce a broader family of time-varying dephasing models and identify a variational resource that is always lower bounded by $\operatorname{fbs}(f)$. Computing this resource reduces to a convex optimization problem, providing a simple way to derive lower bounds for new noise schedules. As applications, we recover the hybrid and IID dephasing bounds and determine the query complexity of unstructured search when the dephasing rate grows over time.
Comments34 pages, LaTex, fixed typos, updated acknowledgement, fixed mathjax in abstract
