Optimization via Quantum Preconditioning

TL;DR

The paper introduces quantum preconditioning for discrete optimization, using QAOA to generate a correlation-based preconditioner that replaces the original weight matrix. By analyzing both infinite-depth and shallow-depth limits, the authors show that the preconditioned problem can dramatically ease local search dynamics and improve convergence of classical solvers such as simulated annealing and the Burer-Monteiro method across random 3-regular graph max-cut, Sherrington-Kirkpatrick spin glasses, and a real-world MPES grid optimization. Practical results from light-cone–based emulations demonstrate a quantum-inspired advantage at shallow depths, with finite preconditioning times still yielding substantial speedups; hardware experiments on superconducting devices illustrate the method's viability and remaining challenges due to noise and sampling. The study highlights the potential for hardware-based quantum utility in optimization, emphasizes the importance of efficient compilation and robust angle optimization, and outlines a roadmap toward constrained problems and error-mitigated quantum preconditioning. Overall, quantum preconditioning emerges as a promising intermediate step toward practical quantum-enhanced optimization, offering concrete pathways to improve classical solver performance today while guiding future hardware developments.

Abstract

State-of-the-art classical optimization solvers set a high bar for quantum computers to deliver utility in this domain. Here, we introduce a quantum preconditioning approach based on the quantum approximate optimization algorithm. It transforms the input problem into a more suitable form for a solver with the level of preconditioning determined by the depth of the quantum circuit. We demonstrate that best-in-class classical heuristics such as simulated annealing and the Burer-Monteiro algorithm can converge more rapidly when given quantum preconditioned input for various problems, including Sherrington-Kirkpatrick spin glasses, random 3-regular graph maximum-cut problems, and a real-world grid energy problem. Accounting for the additional time taken for preconditioning, the benefit offered by shallow circuits translates into a practical quantum-inspired advantage for random 3-regular graph maximum-cut problems through quantum circuit emulations. We investigate why quantum preconditioning makes the problem easier and test an experimental implementation on a superconducting device. We identify challenges and discuss the prospects for a hardware-based quantum advantage in optimization via quantum preconditioning.

Figures (18)

• Figure 1: Average performance for the maximum-cut problem on $N$-variable random $3$-regular graphs via the classical simulated annealing (SA) solver based on the original and quantum-preconditioned problems. (A) Workflow diagram of quantum preconditioning. (B) Average run-time for SA to get to an approximation ratio of $99.9\%$ as a function of $N$. (C) Average approximation ratio versus SA run-time for $N=4,096$. Each data point is averaged over $200$ randomly generated problem instances. Run-times correspond to a 64GB MacBook Pro with an Apple M1 Max chip. Error bars indicate the standard error of the mean.
• Figure 2: Average performance for the maximum-cut problem on $N$-variable random $3$-regular graphs via the classical BM solver based on the original and quantum-preconditioned problems. (A) Average run-time for BM to get to an approximation ratio of $99.9\%$ as a function of $N$. (B) Average approximation ratio versus BM run-time for $N=4,096$. Each data point is averaged over $200$ randomly generated problem instances. Run-times correspond to a 64GB MacBook Pro with an Apple M1 Max chip. Error bars indicate the standard error of the mean.
• Figure 3: Average performance for $N=2,048$ Sherrington-Kirkpatrick spin-glass problems via the SA and Burer-Monteiro (BM) solvers based on the original and quantum-preconditioned $(p=1)$ problems. (A) Average approximation ratio as a function of the number of iterations. (B) Average approximation ratio as a function of the run-time. Each data point is averaged over $200$ randomly generated problem instances. Run-times correspond to a 64GB MacBook Pro with an Apple M1 Max chip. Error bars indicate the standard error of the mean.
• Figure 4: Performance of quantum preconditioning for computing the maximum power exchange section of an energy-grid optimization problem. (A) Representation of the energy-grid optimization problem considered from the state of South Carolina in the United States of America. Vertices ($N=180$) and edges ($n=226$) correspond to buses and lines, respectively. (B) Average approximation ratio as a function of the number of iterations via SA. (C) Average approximation ratio as a function of run-time via SA. (D) Average approximation ratio as a function of the number of iterations via the BM solver. (E) Average approximation ratio as a function of run-time via the BM solver. Run-times are based on a 64GB MacBook Pro with an Apple M1 Max chip. Each data point is averaged over $10^4$ samples. Error bars indicate the standard error of the mean.
• Figure 5: Average time for preconditioning $N$-variable random $3$-regular graph maximum-cut problems using classical state-vector emulations of the QAOA using the light-cone technique at $p=1$ and $p=2$. The maximum preconditioning time budget for an advantage via SA for an approximation ratio of $\alpha=99.9\%$ is computed from Fig. \ref{['fig:sa_preconditioning']}. Run-times correspond to a 64GB MacBook Pro with an Apple M1 Max chip. Error bars indicate the standard error of the mean.