Quantum Solution Framework for Finite-Horizon LQG Control via Block Encodings and QSVT

TL;DR

This paper introduces a quantum algorithm for finite-horizon LQG control that leverages block encodings and QSVT to achieve a potential exponential speedup over classical methods in high-dimensional systems.

Contribution

The paper develops a quantum framework for solving LQG control problems using advanced quantum linear algebra techniques, reducing computational complexity.

Findings

Quantum algorithm scales polylogarithmically with system dimension n.

Total runtime scales linearly with time horizon T.

Potential exponential speedup over classical algorithms.

Abstract

We present a quantum algorithm for solving the finite-horizon discrete-time Linear Quadratic Gaussian (LQG) control problem, which integrates optimal control and state estimation in the presence of stochastic disturbances and noise. Classical approaches to LQG require solving a backward Riccati recursion and a forward Kalman filter, both requiring computationally expensive matrix operations with overall time complexity $\mathcal{O}(T n^3)$, where $n$ is the system dimension and $T$ is the time horizon. While efficient classical solvers exist, especially for small to medium-sized systems, their computational complexity grows rapidly with system dimension. To address this, we reformulate the full LQG pipeline using quantum linear algebra primitives, including block-encoded matrix representations and quantum singular value transformation (QSVT) techniques for matrix inversion and multiplication. We formally analyze the time complexity of each algorithmic component. Under standard assumptions on matrix condition numbers and encoding precision, the total runtime of the quantum LQG algorithm scales polylogarithmically with the system dimension $n$ and linearly with the time horizon $T$, offering an asymptotic quantum speedup over classical methods.