发表机构
School of Mathematical Sciences, Peking University; Beijing International Center for Mathematical Research, Peking University（北京大学数学科学学院; 北京大学北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出两种基于围道积分和高维傅里叶变换的量子算法，用于实现交换矩阵元组的多变量特征值变换，并分析复杂度及混合变体，证明多项式实现无需额外次数依赖。
AI 中文摘要
我们研究在量子计算机上实现多变量矩阵值函数的问题，并提出两种针对成对交换矩阵元组的多变量矩阵特征值变换的量子算法。第一种算法基于多变量围道积分，适用于 $\mathbb{C}^n$ 上的任意全纯函数。第二种算法基于高维傅里叶变换，专为交换厄米矩阵的光滑函数设计。对于这两种算法，我们讨论并分析了利用量子奇异值变换、压缩工具和酉算子线性组合实现其量子实现的复杂度。我们还研究了它们的混合量子-经典变体，这些变体将所需辅助量子比特的数量减少到变量数的对数依赖，但以增加查询复杂度为代价。作为一个应用，我们证明了多变量矩阵多项式可以在没有额外显式次数依赖的情况下实现，仅依赖于多项式的全局性质。
英文摘要
We study the problem of implementing multivariable matrix-valued functions on a quantum computer and propose two quantum algorithms for multivariable matrix eigenvalue transformations acting on tuples of pairwise commuting matrices. The first algorithm is based on multivariable contour integrals and applies to arbitrary holomorphic functions on $\mathbb{C}^n$. The second algorithm is based on high-dimensional Fourier transforms and designed for smooth functions of commuting Hermitian matrices. For both algorithms, we discuss and analyze the complexity of their quantum implementation leveraging quantum singular value transformation, compression gadgets, and linear combination of unitaries. We additionally study their hybrid quantum--classical variants that reduce the number of required ancilla qubits to logarithmic dependence on the number of variables, at the cost of increased query complexity. As an application, we show that multivariable matrix polynomials can be implemented with no additional explicit degree dependence, depending only on global properties of the polynomials.
Comments36 pages including appendix
