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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

Spanish

Desarrollamos algoritmos de sistema de operadores cuánticos fijos para el control óptimo sobre un intervalo de tiempo finito prescrito (horizonte finito), centrándose en la regulación lineal-cuadratica de tiempo continuo (LQR). Este problema de control fundamental a menudo requiere la resolución de una ecuación de matriz Riccati no lineal, que puede ser costosa para sistemas grandes y largos períodos de control en solventes clásicos convencionales. En cambio, trabajamos directamente con las ecuaciones de optimización de estado-costa en un sistema lineal único, reformulando el problema de LQR como un sistema lineal de disparo múltiple. Importante, demostramos una condición-número vinculado para el sistema que es independiente del horizonte, bajo las suposiciones de estabilización y detectibilidad óp

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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