非马尔可夫动力学系统的量子模拟
Quantum simulation of non-Markovian dynamical systems
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中文总结 AI 辅助
本文针对具有记忆特性的非马尔可夫动力学系统，开发了线性沃尔泰拉积分微分方程的高效量子算法，在系统规模上实现指数级加速，扩展了量子计算机可高效模拟的动力学系统范围。
中文摘要 AI 辅助
现有用于模拟动力学系统的量子算法——从哈密顿量模拟到线性和非线性微分方程求解器——均针对马尔可夫动力学，其中系统的未来演化仅取决于其当前状态。本文将研究方向转向开发非马尔可夫动力学系统的量子算法，这类系统的未来演化取决于其过往历史，即具有记忆特性。具体而言，我们针对具有卷积记忆核的线性沃尔泰拉积分微分方程（VIDEs）开发了高效算法，该算法可输出编码某一时间段或特定时间状态描述的量子态。若问题输入存在高效量子电路，我们的算法在系统规模上相较现有经典算法实现了指数级加速。我们针对满足$\textsf{M} < 1$的一般核开发了算法，其中$\textsf{M}$表征记忆项相对于动力学马尔可夫部分耗散的强度。我们补充了$\textsf{M} \geq 1$时一般核VIDEs的下界，表明该问题对一类系统而言是难解的。然而，通过专门研究可在指数上进行简洁分解的结构化核，我们将VIDEs转换为更大的常微分方程组，该过程称为马尔可夫化，即便在$\textsf{M} \geq 1$时也能开发高效量子算法。作为该整体框架的应用，我们讨论了开放量子系统和流体动力学中使用的Mori-Zwanzig形式。总体而言，我们的结果扩展了量子计算机可高效模拟的动力学系统范围。
英文摘要
Existing quantum algorithms for simulating dynamical systems -- from Hamiltonian simulation to linear and nonlinear differential equations solvers -- simulate Markovian dynamics, in which the system's future evolution depends solely on its current state. We turn our attention to developing quantum algorithms for non-Markovian dynamical systems where the system's future evolution depends on its past history and thus has memory. Specifically, we develop efficient algorithms for linear Volterra integro-differential equations (VIDEs) with a convolution memory kernel that output a quantum state encoding the state description over a time interval or at a particular time. Given efficient circuits for the problem inputs, our algorithms achieve an exponential speedup in system size over existing classical algorithms. We develop an algorithm for general kernels assuming that $\textsf{M} < 1$, where $\textsf{M}$ characterizes the strength of the memory term relative to the dissipation of the Markovian part of the dynamics. We complement this with lower bounds for general-kernel VIDEs when $\textsf{M} \geq 1$, showing that the problem becomes intractable for a family of systems. However, by specializing to structured kernels which admit concise decompositions over exponentials, we develop efficient quantum algorithms even when $\textsf M \geq 1$ by converting the VIDE into a larger set of ODEs, a procedure which we call Markovianization. As an application of the overall framework, we discuss the Mori-Zwanzig formalism used in open quantum systems and fluid dynamics. Overall, our results expand the range of dynamical systems that quantum computers can simulate efficiently.
