黎曼流形上的量子模拟
Quantum Simulation on Riemannian Manifolds
- Global Technology Applied Research, JPMorganChase（摩根大通全球技术应用研究）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出黎曼流形上薛定谔方程的量子模拟算法，涵盖全局谱方法与多坐标卡局部方法，并应用于优化与物理模拟，包括量子哈密顿下降的推广及非线性西格玛模型的模拟。
AI中文摘要：
我们研究了黎曼流形上薛定谔方程的量子模拟算法，其中动能算子由对应于度量的拉普拉斯-贝尔特拉米算子定义。我们的第一类算法基于全局谱方法，该方法依赖于识别到拉普拉斯-贝尔特拉米算子特征基的高效变换。我们利用此方法为以下情形提供了显式、高效的量子模拟算法：具有标准度量的环面和球面上的黎曼薛定谔方程，具有赖特-费舍尔度量的单纯形，截断正象限及其可逆仿射像（带对数障碍黑塞度量），以及由达菲映射诱导度量的ℓ_p球。我们的第二类算法基于在多个坐标卡上对局部谱方法的相干模拟，原则上适用于任何紧流形。我们首先在连续情形下分析该算法，并推导出多项式谱截断充分的条件。我们还对常维流形张量积的多项式谱截断进行了离散化分析。最后，我们考虑了这些方法在优化和物理模拟中的应用。对于优化，我们提供了包括测地凸函数的量子哈密顿下降的推广和收敛性分析在内的结果，该分析导出了球面和单纯形上的显式算法，以及实空间绝热算法的黎曼推广。对于物理模拟，我们表明我们的算法可以模拟某些空间离散化场论，包括非线性西格玛模型的一种变体。
英文摘要：
We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and $\ell_p$ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.
