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IEEE International Conference on Quantum Computing and Engineering (QCE)

Adiabatic Quantum Linear Optimal Control for Discrete Time Dynamical Systems

Mayank Jha, Prasanna Date

Published Aug 30, 2025• 0 citations
Publisher

Abstract

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$O\left((N m)^{3}\right)$for horizon length$N$and control dimension$m$. 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$N$. 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$N$. 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.

DOI
10.1109/qce65121.2025.10341

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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 asO((Nm)3)O\left((N m)^{3}\right)O((Nm)3)for horizon lengthNNNand control dimensionmmm. 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 onNNN. 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 ofNNN. 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.
This paper introduces an innovative approach to explore the capabilities of Quantum Annealing (QA) for trajectory optimization in dynamic systems. The proposed method involves transforming trajectory optimization problems into equivalent binary optimization problems using Quadratic Unconstrained Binary Optimization (QUBO) representation. The procedure is general and adaptable, making it applicable to a wide range of optimal control problems that entail the satisfaction of dynamic, boundary, and path constraints. Specifically, the trajectory is discretized and approximated using polynomials. In contrast to the conventional approach of determining the polynomial degree (n) solely based on the number of boundary conditions, a specific factor is introduced in our method to augment the polynomial degree. As a result, the ultimate polynomial degree is calculated as a composite of two components: n = l + (m−1), where m denotes the count of boundary conditions and l signifies the number of independent variables. By leveraging inverse dynamics, the control required to follow the approximated trajectory can be determined as a linear function of independent variables l. As a result, the optimization function, which is represented by the integral of the square of the control, can be formulated as a QUBO problem and the QA is employed to find the optimal binary solutions.
Quantum annealers can solve QUBO problems efficiently but struggle with continuous optimization tasks like regression due to their discrete nature. We introduce Quadratic Continuous Quantum Optimization (QCQO), an anytime algorithm that approximates solutions to unconstrained quadratic programs via a sequence of QUBO instances. Rather than encoding real variables as binary vectors, QCQO implicitly represents them using continuous QUBO weights and iteratively refines the solution by summing sampled vectors. This allows flexible control over the number of binary variables and adapts well to hardware constraints. We prove convergence properties, introduce a step size adaptation scheme, and validate the method on linear regression. Experiments with simulated and real quantum annealers show that QCQO achieves accurate results with fewer qubits, though convergence slows on noisy hardware. Our approach enables quantum annealing to address a wider class of continuous problems.
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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 complexityO(Tn3)\mathscr{O}\left(T n^{3}\right)O(Tn3), wherennnis the system dimension andTTTis 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 dimensionnnnand linearly with the time horizonTTT, offering an asymptotic quantum speedup over classical methods.
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.
Abstract We benchmark classical, simulated-quantum, hybrid, and direct quantum-annealing workflows on a structured binary control problem formulated as a quadratic unconstrained binary optimization (QUBO) model. The benchmark is motivated by greenhouse heater scheduling, where the horizon (H)(H)(H) denotes the number of hourly binary control decisions, but the focus is on solver behavior for a physically decoded control-QUBO formulation. For the main one-day instance ((H=24)(H=24)(H=24)), all solver outputs are decoded back into heater schedules and evaluated using the same simulator-side objective and feasibility criterion. Classical simulated annealing and path-integral simulated quantum annealing produce feasible near-optimal solutions in all repetitions, with best objectives close to the exact optimum. In contrast, the tested D-Wave Leap Hybrid binary quadratic model workflow is less reliable and does not outperform the classical baselines under 15--60 s requested time limits. Direct D-Wave quantum processing unit execution on reduced instances remains feasible in all runs and recovers the exact optimum for (H=10)(H=10)(H=10) and (H=12)(H=12)(H=12), but the exact-hit rate decreases with increasing horizon. The results do not indicate quantum advantage, but provide a reproducible benchmark for evaluating annealing-based natural and quantum computing workflows on structured control QUBOs. MSC Classification: 90C27 , 90C59 , 81P68 , 93C55