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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

Korean

우리는 정해진 제한된 시간 간격 (제한된 지평)에 걸쳐 최적의 제어를 위해 정해진 숫자 시스템 알고리즘을 개발하여 지속 시간 선형 사각형 규제 (LQR)에 초점을 맞추고 있습니다. 이 기본 제어 문제는 종종 대형 시스템과 일반적인 고전 솔버에서 장기간의 제어 비용이 들 수있는 비선형 매트릭스 Riccati 방정식을 해결하는 것을 필요로합니다. 대신에 우리는 단일 선형 시스템에 최적 상태와 최종 오류 조건을 직접 통합하여 LQR 문제를 다중 샷 선형 시스템으로 재구성합니다. 중요한 것은, 우리는 경계에 독립적 인 시스템에 대한 조건 숫자 결합을 증명합니다. 안정성과 감지 가능성 가정에 따라 최적의 체계

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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