发表机构
Center for Quantum Computer Science, Faculty of Computing, University of Latvia; Alfréd Rényi Institute of Mathematics, Budapest, Hungary（拉脱维亚大学计算学院量子计算机科学中心; 阿尔弗雷德·雷尼数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在查询模型中建立混合量子-经典算法的紧界，证明相位估计、振幅估计、搜索及AND-OR树问题的最优查询复杂度，并发展通用下界证明方法。
AI 中文摘要
我们研究了查询模型中的混合量子-经典算法。计算过程由量子子程序组成，每个子程序最多进行 $q$ 次预言机查询；在子程序之间，所有量子比特被测量并丢弃。我们证明了该模型中多个问题的匹配上下界：(i)（无偏）相位估计达到精度 $\epsilon$ 需要 $\Theta(\frac{1}{q \epsilon^2})$ 次查询；(ii)（无偏）振幅估计达到精度 $\epsilon$ 需要 $\Theta(\frac{p(1-p)}{q \epsilon^2})$ 次查询；(iii) 在 $N$ 个项目中搜索需要 $\Theta(\frac{N}{q})$ 次查询；(iv) 两级 AND-OR 树（$m$ 个 OR 的 AND，每个 OR 有 $n$ 个输入）需要 $\Theta(\frac{nm}{q})$ 次查询。所有这些界在常数因子内都是最优的，适用于从 1（对应经典复杂度）到 $Q(f)$（相应问题的无限制量子查询复杂度）的所有 $q$ 值。我们还推导了一个用于区分由状态制备酉算子指定的两个分布的混合下界；该界在对数因子内是紧的。除了具体的下界，一个重要贡献是开发了证明混合量子算法下界的方法（迄今为止这些方法非常特设）。前两个界遵循一个共同框架：我们分析测量记录上的概率分布，并界定一个捕捉其可区分性的进度度量。AND-OR 界需要更精细的论证，结合两个分别跟踪算法在经典和量子形式下可用信息的进度度量。
英文摘要
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most $q$ oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision $ε$ requires $Θ(\frac{1}{q ε^2})$ queries; (ii) (Unbiased) Amplitude estimation up to a precision $ε$ requires $Θ(\frac{p(1-p)}{q ε^2})$ queries; (iii) Search among $N$ items requires $Θ(\frac{N}{q})$ queries; (iv) Two level AND-OR tree (AND of $m$ ORs, with $n$ inputs to each OR) requires $Θ(\frac{nm}{q})$ queries. All of these bounds are optimal up to a constant factor, for all $q$ from 1, corresponding to the classical complexity, to $Q(f)$, the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.
Comments19 pages
