发表机构
Monash University; Universitas Kristen Immanuel（蒙纳士大学; 伊曼纽尔基督大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明，在稀疏预言机模型下，制备 Hodge Laplacian 线性系统归一化最小范数解的量子态是 BQP-困难的，且该问题为 BQP-完备，为高阶网络分析中的量子优势提供了最坏情况复杂性基础。
AI 中文摘要
高阶网络（HONs）表示对象群体之间的交互，支持研究成对模型可能遗漏的集体行为。在这些网络的单纯复形模型中，Hodge Laplacian 线性系统提供了一个通用的数学分析框架，例如用于解决统计排名问题和研究耦合振荡器系统中的长期稳定性。与边、三角形和高维单纯形相关的大量变量可能使这些系统的求解成本高昂。最近的量子算法解决了这一瓶颈。然而，与特定经典算法的比较并不能决定性地表明求解这些线性系统问题是否建立了可证明量子优势的量子复杂性基础。我们证明，在指定的稀疏预言机模型和参数承诺下，制备编码 Hodge Laplacian 线性系统的归一化最小范数解的量子态是 $\u000cmathsf{BQP}$-困难的。我们的归约序列将任意多项式时间有界误差量子计算映射到此类线性系统。每个归约都允许高效恢复前一个线性系统的归一化最小范数解。结合用于从近似解态解决相关决策问题的高效量子算法，我们表明该问题是 $\u000cmathsf{BQP}$-完备的。这些结果为评估 HON 分析中的量子优势提供了最坏情况复杂性基础，并表明量子线性系统问题（QLSP）的 $\u000cmathsf{BQP}$-困难性即使在矩阵被限制为 Hodge Laplacian 时仍然存在。
英文摘要
Higher-order networks (HONs) represent interactions among groups of objects, supporting the study of collective behaviour that pairwise models can miss. In simplicial models of these networks, Hodge Laplacian linear systems provide a common mathematical framework for analysis, for example, for solving statistical ranking problems and investigating long-term stability in coupled oscillator systems. The large number of variables associated with edges, triangles, and higher-dimensional simplices can make these systems costly to solve. Recent quantum algorithms address this bottleneck. However, comparisons with particular classical algorithms do not decision output whether solving these linear system problems establishes a quantum complexity foundation for provable quantum advantage. We prove that preparing a quantum state encoding the normalized minimum-norm solution of Hodge Laplacian linear systems is $\mathsf{BQP}$-hard under the specified sparse oracle model and parameter promises. Our sequence of reductions maps an arbitrary polynomial-time bounded-error quantum computation to such linear systems. Each reduction allows efficient recovery of the normalized minimum-norm solution of the preceding linear system. Together with an efficient quantum algorithm for the associated decision problem from the approximate solution state, we show that this problem is $\mathsf{BQP}$-complete. These results provide a worst-case complexity foundation for evaluating quantum advantage in HON analysis and show that the $\mathsf{BQP}$-hardness of the quantum linear system problem (QLSP) persists even when the matrices are restricted to Hodge Laplacians.
