Adiabatic Quantum Linear Optimal Control for Discrete Time Dynamical Systems

This paper presents a novel approach for solving finite-horizon linear optimal control based on block-Toeplitz least-squares problem, by recasting the latter as Quadratic Unconstrained Binary Optimization (QUBO) suitable for adiabatic quantum computing (AQC). Classical block-Toeplitz leastsquares problem, scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O\left((N m)^{3}\right)$</tex> for horizon length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> and control dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex>. We demonstrate that this Toeplitzstructured least-squares cost can be transformed into a QUBO formulation by introducing binary precision vectors that encode each continuous control parameter into finite number of bits. We establish the general case in mathematically rigorous manner wherein the total number of binary decision variables remain dependent on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex>. To respect current quantum annealer capabilities, we propose a basis-function parametrization that approximates the full control sequence with a small set of basis coefficients, reducing the total number of binary-variables and rendering the latter independent of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex>. Through simulation study, we show that, for large datasets, the QUBO pipeline, comprising of hardware-constant anneal step outperforms classical least square based solvers by several factors whilst demonstrating acceptable accuracy.