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Quantum algorithms for optimal control: large-scale finite-horizon linear-quadratic regulation

Xiajie Huang, Cristina Álvarez Yuste, Nikita Guseynov, Nana Liu, Javier Gonzalez-Conde · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.26843 · cited by 0 works (OpenAlex) · open-access PDF

Chinese

我们专注于连续时间线性平方调节(LQR)的量子操作系统的固定算法,以便在规定的有限时间间隔(finite horizon)上进行最佳控制,专注于连续时间线性平方调节(LQR)。这个基本控制问题通常需要解决非线性矩阵Riccati方程式,这可能对大型系统和传统经典解析器的长时间控制成本高昂。相反,我们直接与状态边际优化方程式合作,将LQR问题改编为多射线性系统。重要的是,我们证明了对系统的条件数约束,而这与地平线无关,在稳定性和可检测性假设下,我们可以对系统的条件和最终错误进行直接处理。通过将初始条件和错误条件纳入一个单一线性系统,我们

Machine translation (Meta M2M100); the English original follows.

Abstract

We develop quantum algorithms for optimal control over a prescribed finite time interval (finite horizon), focusing on continuous-time linear-quadratic regulation (LQR). This foundational control problem often requires solving a nonlinear matrix Riccati equation, which can be costly for large systems and long control durations on conventional classical solvers. We instead work directly with the state-costate optimality equations, reformulating the LQR problem as a multiple-shooting linear system. Importantly, we prove a condition-number bound for the system that is independent of the horizon, under stabilizability and detectability assumptions. By incorporating the initial and final conditions into a single linear system, we can solve the subsequent linear system more efficiently with a quantum linear systems subroutine. For a given initial state, our quantum algorithms return classical estimates of the optimal cost and the optimal control inputs at a selected time, without constructing a Riccati solution operator. Given sparse-matrix block-encodings, an initial-state preparation oracle and a fixed number of control inputs, the query and gate complexities scale polylogarithmically with system size and control duration, provided that sparsity, relevant norms, and stability parameters are bounded. The dependence on the target error is inverse-linear up to logarithmic factors. We establish the required bounds uniformly in system size and duration for a family of damped mass-spring chains with two endpoint controls. Numerical experiments support the analysis.

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Topics: Mathematics, Optimal control, Algebraic Riccati equation, Quantum, Logarithm, Linear system

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