发表机构
University of the Basque Country, UPV/EHU; EHU Quantum Center, University of the Basque Country UPV/EHU; IBM Quantum, IBM Research Europe; Quantum Mads; Basque Center for Applied Mathematics (BCAM); IKERBASQUE, Basque Foundation for Science（巴斯克大学; 巴斯克大学EHU量子中心; IBM欧洲研究院量子部门; Quantum Mads; 巴斯克应用数学中心; 伊克尔巴斯基克科学基金会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出量子算法近似带状Toeplitz矩阵指数，通过LCU块编码与QFT-Trotter公式，应用于偏微分方程离散热方程传播子。
AI 中文摘要
我们提出了量子算法来近似带状Toeplitz矩阵的指数。这些矩阵在许多与偏微分方程相关的问题中具有核心重要性，但它们的量子实现受到其可能较大范数的阻碍。通过直接构造指数，我们可以规避这一限制。通过将下/上移位算子与由量子傅里叶变换（QFT）对角化的循环矩阵和斜循环矩阵生成元相关联，我们构建了（i）基于线性组合酉算子（LCU）的带状Toeplitz矩阵块编码，（ii）循环矩阵特征相位的高效、可控截断的Pauli字符串分解，并带有闭式误差界，以及（iii）专门的QFT-Trotter乘积公式。作为应用，我们使用频率截断，构建了具有周期、Dirichlet和Neumann边界条件的离散化热方程传播子的块编码。
英文摘要
We present quantum algorithms to approximate the exponential of banded Toeplitz matrices. These matrices have central importance in many PDE related problems, but their quantum implementation is hindered by their possibly large norms. Constructing directly the exponential we can circumvent this limitation. By relating the lower/upper shift operators to circulant and skewcirculant generators, which are diagonalised by the QFT, we construct (i) an LCU-based block encoding of banded Toeplitz matrices, (ii) an efficient, controllably truncated Pauli-string decomposition of the circulant eigenphases with a closed-form error bound, and (iii) a specialized QFT-Trotter product formula. As an application we build a block encoding of the propagator of the discretised heat equation with periodic, Dirichlet and Neumann boundary conditions, using a frequency cutoff.
