← 研踪 PaperTrailarXiv:2611.09809 · v1
Beyond the $n/2$ Threshold: Quantum Query Complexity for Computing Hamming Weight and Symmetric Functions
首次提交 2026-11-07 01:58
版本更新 2026-11-07 01:58
雷达观测 2026-11-09 13:22
摘要
The oracle-interrogation algorithm [van Dam, FOCS '98] computes every $n$-bit Boolean function with bounded error using $n/2+O(\sqrt n)$ quantum queries. In contrast, parity can be computed exactly using $\lceil n/2\rceil$ queries, but whether an additive overhead beyond $n/2$ is necessary for a single Boolean output has remained open. In this work, we construct an explicit family of symmetric Boolean functions requiring $n/2+Ω(n^{1/3})$ quantum queries at a fixed positive constant error. We also give an algorithm that recovers the exact Hamming weight using $n/2+O(n^{1/3})$ quantum queries with bounded error, and therefore computes every symmetric Boolean function within the same query bound. Together, these results determine the optimal additive order for Hamming weight recovery and for the worst case over symmetric Boolean functions at sufficiently small constant error.
作者备注：57 pages, 2 figures
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v1 · 2026-11-07 01:58 · 最新观测版本
Beyond the $n/2$ Threshold: Quantum Query Complexity for Computing Hamming Weight and Symmetric Functions
The oracle-interrogation algorithm [van Dam, FOCS '98] computes every $n$-bit Boolean function with bounded error using $n/2+O(\sqrt n)$ quantum queries. In contrast, parity can be computed exactly using $\lceil n/2\rceil$ queries, but whether an additive overhead beyond $n/2$ is necessary for a single Boolean output has remained open. In this work, we construct an explicit family of symmetric Boolean functions requiring $n/2+Ω(n^{1/3})$ quantum queries at a fixed positive constant error. We also give an algorithm that recovers the exact Hamming weight using $n/2+O(n^{1/3})$ quantum queries with bounded error, and therefore computes every symmetric Boolean function within the same query bound. Together, these results determine the optimal additive order for Hamming weight recovery and for the worst case over symmetric Boolean functions at sufficiently small constant error.
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