论双Koopman-von Neumann嵌入在量子计算机上求解保守非线性常微分方程的数值局限性
On the numerical limitations of dual Koopman von Neumann embeddings for solving conservative nonlinear ordinary differential equations on quantum computers
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中文总结 AI 辅助
本文提出基于Koopman-von Neumann嵌入的量子算法求解非线性常微分方程，推导Ehrenfest-Reynolds数稳定性判据，数值验证误差增长，为初始高斯分布宽度选择提供指导。
中文摘要 AI 辅助
在量子计算机上模拟非线性常微分方程 inherently 具有挑战性，因为量子门是量子比特态上的线性算子。本文提出了一种基于Koopman-von Neumann（KvN）算子的算法，用于在量子计算机上求解非线性常微分方程，该算法克服了量子操作的固有局限性。在该方法中，Liouville概率密度被嵌入到波函数中，其演化由与Koopman算子对偶的算子控制。对应于非线性微分方程的轨迹通过量子测量评估的Koopman可观测量平均值来重建。我们特别评估了在该Liouville嵌入框架内求解非线性方程的计算局限性。推导了一个Ehrenfest型估计，将重建误差与输运密度的协方差联系起来，突出了线性拉伸与Hessian诱导效应之间的竞争。这导致了一个关键稳定性判据，该判据依赖于局部Ehrenfest-Reynolds数。我们讨论了测量诱导误差的影响，包括Hadamard测试采样、振幅估计以及概率性Grover型推断中协方差引起的偏差对轨迹评估的影响。量子计算精度的极限边界通过Lotka-Volterra系统和四次振子进行了数值验证。我们观察到当Ehrenfest-Reynolds数接近预测阈值时，计算误差迅速增长。我们分析得出的边界为选择与非线性常微分方程系统相关的初始高斯概率分布的宽度提供了关键指导。
英文摘要
The simulation of nonlinear ordinary differential equations on quantum computers is inherently challenging, as quantum gates are linear operators on qubit states. In this paper, we put forth a Koopman-von Neumann (KvN) operator based algorithm for solving nonlinear ordinary differential equations on a quantum computer which overcomes the innate limitations of quantum operations. In this approach, a Liouville probability density is embedded into a wavefunction, the evolution of which is governed by an operator dual to the Koopman operator. The trajectories corresponding to the nonlinear differential equations are reconstructed from the average value of Koopman observables evaluated through quantum measurements. We specifically evaluate the computational limitations of solving nonlinear equations within this Liouville embedding framework. An Ehrenfest-type estimate is derived that relates the reconstruction error to the covariance of the transported density, highlighting the competition between linear stretching and Hessian-induced folding.This leads to a key stability criterion, which depends on the local Ehrenfest-Reynolds number. We discuss the effects of measurement-induced errors including those due to Hadamard-test sampling, amplitude estimation, and bias due to covariance in probabilistic Grover-type inference on the evaluation of trajectories. The limiting bounds on the accuracy of quantum computations are numerically verified for the Lotka-Volterra system and for the quartic oscillator. We observe a rapid growth in computational errors when the Ehrenfest-Reynolds number approaches the predicted threshold. The bounds resulting from our analysis provide key guidelines for selecting the width of the initial Gaussian probability distribution associated with a system of nonlinear ordinary differential equations.
发表机构
- Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）
- Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LUX（索邦大学、巴黎天文台、巴黎文理研究大学、法国国家科学研究中心、LUX）
- CERFACS（欧洲流体计算与应用研究中心）
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