发表机构
Beijing International Center for Mathematical Research, Peking University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Alfréd Rényi Institute of Mathematics; School of Mathematical Sciences, Peking University（北京大学数学中心; 中国科学院数学与系统科学研究院; 阿尔弗雷德·雷尼数学研究所; 北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于换能器的LCU框架，将最昂贵酉算子成本降为加权平均查询复杂度，并应用于稀疏矩阵块编码，实现近最优的哈密顿量模拟与量子线性系统算法。
AI 中文摘要
酉算子线性组合（LCU）是量子算法中的一个基本原语，其成本通常由组合中出现的最昂贵的酉算子决定。我们开发了一个基于换能器的LCU框架，当组成酉算子共享对一组原始预言机的访问时，该框架将这种最坏情况依赖性降低为加权平均查询复杂度。考虑$A=\sum_j c_j U_j$，其中$c_j>0$且每个酉算子$U_j$可以使用$C_j$次原始查询实现。给定上界$a\geq \\|A\\|$，我们的算法实现了$A/\alpha$的块编码，重缩放因子$\alpha=\mathcal{O}(a)$，使用$\widetilde{\mathcal{O}} (C_{\max}+\overline{C} {\lambda}/{a} )$次原始查询，其中$\lambda=\sum_j c_j$，$C_{\max}=\max_j C_j$，$\overline{C}= {\sum_j c_j C_j}/{\lambda}$。相比之下，标准LCU构造需要$\widetilde{\mathcal{O}} (C_{\max} {\lambda}/{a} )$次原始查询。因此，当昂贵的酉算子权重较小且$\lambda/a$较大时，改进可能是显著的。作为应用，我们获得了稀疏矩阵的改进块编码，从而得到了稀疏哈密顿量模拟和量子线性系统的量子算法，在所有相关参数上具有接近最优的查询复杂度，直至多对数因子。我们的主要技术是Belovs、Jeffery和Yolcu开发的换能器框架[Quantum, 8:1444 (2024)]。在这里，我们开发了一个针对块编码的补充算子级理论。我们识别出解析范数$K$作为换能器实现的关键复杂度度量，除了现有的催化剂复杂度之外。我们表明，换能器作用可以转换为$\epsilon$-近似的块编码，仅使用对换能器的$\mathcal{O}\left(K \log(1/\epsilon)\right)$次查询，将精度依赖性从多项式改进为对数。
英文摘要
Linear combination of unitaries (LCU) is a fundamental primitive in quantum algorithms, whose cost is typically governed by the most expensive unitary appearing in the combination. We develop a transducer-based LCU framework that reduces this worst-case dependence to a weighted average query complexity, when the constituent unitaries share access to a common set of primitive oracles. Consider $A=\sum_j c_j U_j$, where $c_j>0$ and each unitary $U_j$ can be implemented using $C_j$ primitive queries. Given an upper bound $a\geq \|A\|$, our algorithm implements a block-encoding of $A/α$ with rescaling factor $α=\mathcal{O}(a)$, using $\widetilde{\mathcal{O}} (C_{\max}+\overline{C} λ/{a} )$ primitive queries, where $λ=\sum_j c_j$, $C_{\max}=\max_j C_j$, and $\overline{C}= {\sum_j c_j C_j}/λ$. By comparison, the standard LCU construction requires $\widetilde{\mathcal{O}} (C_{\max} λ/{a} )$ primitive queries. The improvement can therefore be substantial when costly unitaries have small weights and $λ/a$ is large. As applications, we obtain improved block-encodings of sparse matrices, leading to quantum algorithms for sparse Hamiltonian simulation and quantum linear systems with near-optimal query complexity up to polylogarithmic factors in all relevant parameters. Our main technique is the transducer framework developed by Belovs, Jeffery, and Yolcu [Quantum, 8:1444 (2024)]. Here we develop a complementary operator-level theory tailored to block-encodings. We identify the resolvent norm $K$ as a key complexity measure for transducer implementation besides the existing catalyst complexity. We show that the transducer action can be converted into an $ε$-approximate block-encoding using only $\mathcal{O}\left(K \log(1/ε)\right)$ queries to the transducer, improving the precision dependence from polynomial to logarithmic.
