发表机构
University of Edinburgh; Institute of Software, Chinese Academy of Sciences; State Key Laboratory for Novel Software Technology, Nanjing University; Hefei National Laboratory（爱丁堡大学; 中国科学院软件研究所; 南京大学软件新技术国家重点实验室; 合肥国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究量子 junta 测试的查询复杂度，在仅前向模型下给出酉算子和信道的紧界，确立最优复杂度为 Θ(k/(ε²log k))，并揭示其与经典支持大小测试的深层联系。
AI 中文摘要
我们研究了量子 junta 测试的查询复杂度。给定一个未知的 $n$ 量子比特酉算子或量子信道，目标是判定它是否在至多 $k$ 个量子比特上非平凡作用，还是与每个这样的操作 $\varepsilon$-远。对于酉算子，我们考虑仅前向模型，其中测试者可以访问 $U$，但不能访问 $U^\dagger$ 或受控-$U$。我们给出了酉算子和信道 junta 测试的紧界。对于足够大的 $k$，$n \ge 2k$，以及广泛的 $\varepsilon$ 范围，两个问题的查询复杂度均为 $\Theta\left(k/(\varepsilon^2\log k)\right)$。我们的上界通过使用乘积输入和单量子比特测量的简单非自适应测试器实现。对于 junta 信道，这填补了 Bao 和 Yao（COLT 23, TPAMI 2025）在常数 $\varepsilon$ 下 $\widetilde{O}(k)$ 上界与 $\widetilde{\Omega}(\sqrt{k})$ 下界之间的差距，确立了最优的 $\Theta(k/\log k)$ 查询复杂度。两个界的核心是量子 junta 测试与经典支持大小测试之间的强联系。对于上界，我们将量子 junta 测试归约为随机子集上分布的广义支持大小问题，并开发了一个具有改进样本复杂度的新测试器。对于下界，我们反向进行，通过相位掩蔽硬系综将经典支持大小测试归约为量子 junta 测试，并表明前向量子查询在平均意义上可以被精确经典化。这些归约共同将支持大小测试确定为量子 junta 测试的经典核心，并产生匹配的下界。在常数 $\varepsilon$ 下，它们还给出了 $\Theta(k/\log k)$ 前向查询与 $\widetilde{O}(\sqrt{k})$ 逆访问查询之间的分离。
英文摘要
We study the query complexity of testing quantum juntas. Given an unknown $n$-qubit unitary or quantum channel, the goal is to decide whether it acts nontrivially on at most $k$ qubits or is $\varepsilon$-far from every such operation. For unitaries, we consider the forward-only model, where the tester has access to $U$, but not to $U^\dagger$ or controlled-$U$. We give tight bounds for both unitary and channel junta testing. For sufficiently large $k$, $n \ge 2k$, and a broad range of $\varepsilon$, both problems have query complexity $Θ\left(k/(\varepsilon^2\log k)\right)$. Our upper bounds are achieved by simple nonadaptive testers using product inputs and single-qubit measurements. For junta channels, this closes the gap by Bao and Yao (COLT 23, TPAMI 2025) between an $\widetilde{O}(k)$ upper bound and a $\widetildeΩ(\sqrt{k})$ lower bound at constant $\varepsilon$, establishing the optimal $Θ(k/\log k)$ query complexity. At the heart of both bounds is a strong connection between quantum junta testing and classical support-size testing. For the upper bounds, we reduce quantum junta testing to a generalized support-size problem for distributions over random subsets, and develop a new tester with improved sample complexity. For the lower bounds, we proceed in the reverse direction, reducing classical support-size testing to quantum junta testing through a phase-masked hard ensemble and showing that forward quantum queries can be exactly classicalized on average. Together, these reductions identify support-size testing as the classical core of quantum junta testing and yield matching lower bounds. At constant $\varepsilon$, they also give a separation between $Θ(k/\log k)$ forward queries and $\widetilde{O}(\sqrt{k})$ queries with inverse access.
Comments57 pages, 1 table, 3 algorithms
