A New Hybrid Quantum-Classical Algorithm for Solving the Unit Commitment Problem

TL;DR

This paper tackles the unit commitment problem, a large-scale mixed-integer optimization, by introducing a hybrid quantum-classical framework that combines a variational quantum algorithm (VQA) with a Benders-type decomposition. The VQA serves as a quantum sieve to generate a compact set of promising unit-commitment vectors via a QUBO formulation and a layered Butterfly ansatz, while a classical SLSQP refinement solves residual dispatch problems for fixed commitments. The approach achieves exact optimality for a 3-unit instance and close-to-optimal results for 10- and 26-unit cases, with hardware demonstrations on IonQ Forte showing mean approximation errors around 3%. The work demonstrates a viable pathway for integrating quantum acceleration into grid optimization and outlines concrete avenues for scaling and constraint handling in future developments.

Abstract

Solving problems related to planning and operations of large-scale power systems is challenging on classical computers due to their inherent nature as mixed-integer and nonlinear problems. Quantum computing provides new avenues to approach these problems. We develop a hybrid quantum-classical algorithm for the Unit Commitment (UC) problem in power systems which aims at minimizing the total cost while optimally allocating generating units to meet the hourly demand of the power loads. The hybrid algorithm combines a variational quantum algorithm (VQA) with a classical Bender's type heuristic. The resulting algorithm computes approximate solutions to UC in three stages: i) a collection of UC vectors capable meeting the power demand with lowest possible operating costs is generated based on VQA; ii) a classical sequential least squares programming (SLSQP) routine is leveraged to find the optimal power level corresponding to a predetermined number of candidate vectors; iii) in the last stage, the approximate solution of UC along with generating units power level combination is given. To demonstrate the effectiveness of the presented method, three different systems with 3 generating units, 10 generating units, and 26 generating units were tested for different time periods. In addition, convergence of the hybrid quantum-classical algorithm for select time periods is proven out on IonQ's Forte system.

Figures (8)

• Figure 1: Illustration of our layered alternating ansatz $\ket{\psi(\boldsymbol{\theta})}$ on $N = 6$ qubits, with $2$ layers, and $\boldsymbol{\theta} = (\boldsymbol{\gamma}^{(1)}, \beta_1, \boldsymbol{\gamma}^{(2)}, \beta_2)$. Here $U_E$ denotes the entangling block and $U_M$ denotes the mixer. The entangling block has the structure of the Butterfly Ansatz, as in Cherrat2024quantumvision, constructed with parameteric $ZY$ gates, and the mixer is constructed so its ground state $\ket{\mathbf{u}^*}_{\mathrm{relax}}$ encodes the solution to the semi-definite relaxation of Problem \ref{['eq:lp-qubo']}. We note that each $\boldsymbol{\gamma}^{(\ell)}$ is a parameter vector with $O(\log N)$ parameters, one for each "layer" in the Butterfly ansatz.
• Figure 2: (Left panel) Mean approximation error across $24$ time periods in our $10$-unit problem. (Right panel) Mean number of COBYLA iterations required for convergence to a tolerance of $10^{-6}$ across $24$ time periods in our $10$-unit UC instance. In both cases, the results are averaged over $7$ independent trials. Each trial consisted of a noiseless quantum simulation of the VQA described in Section \ref{['sec:methodology']}, with $\lambda = 450,000$ and solving at most $128$ residual QPs for each hourly problem. Combined, these figures illustrate the tradeoff between the total cost of the ansatz parameter optimization routine and the achievable approximation error.
• Figure 3: (Left panel) Mean approximation error across $24$ time periods in our $26$-unit problem. (Right panel) Mean number of COBYLA iterations required for convergence to a tolerance of $10^{-6}$ across $24$ time periods in our $26$-unit UC instance. In both cases, the results are averaged over $3$ independent trials. Each trial consisted of a noiseless quantum simulation of the VQA described in Section \ref{['sec:methodology']}, with $\lambda = 700,000$ and solving at most $128$ residual QPs for each hourly problem. Combined, these figures illustrate the tradeoff between the total cost of the ansatz parameter optimization routine and the achievable approximation error.
• Figure 4: Approximation error obtained when using the quantum simulator and IonQ Forte to sample the optimized quantum state $\ket{\psi(\boldsymbol{\theta^*})}$, described in Section \ref{['sec:methodology']}, using optimized ansatz parameters computed by simulating our VQA. For each time period in our $26$-unit problem, we measure $2,000$ samples of the optimized state on each backend and then we solve at most $128$ residual QPs corresponding to the sampled feasible assignment vectors with the lowest minimum operating cost. Here $\ket{\psi}$ denotes our layered ansatz with $1$ gate layer. The average approximation error across time periods when using simulated samples is $3.201\%$, and it is $3.088\%$ when using samples measured by IonQ Forte.
• Figure 5: Convergence of our VQA running on IonQ Forte as it approximately solves an hourly problem with $26$ generating units. For this experiment we used our layered ansatz with a single layer and $512$ shots per iteration. All parameters were initialized to zero.