Extending quantum annealing to continuous domains: a hybrid method for quadratic programming

TL;DR

This work addresses extending quantum annealing to continuous optimization by embedding QA within a simulated annealing loop to guide descent in box-constrained quadratic programs. The method, QESA, uses QA to solve Ising subproblems that determine discrete search directions, updating the continuous variable via $x' = x + k s$ with projection to $[-1,1]^n$ and Metropolis acceptance. Key contributions include the Ising formulation for direction optimization, boundary-based initialization, and an extensive empirical evaluation showing that QESA outperforms classical baselines and can surpass constrained solvers like Gurobi under time limits on larger, ill-conditioned instances. The results demonstrate that quantum-guided search directions provide a significant accuracy advantage while maintaining practical runtimes, highlighting QESA’s potential as a scalable hybrid optimization framework as QA hardware matures.

Abstract

We propose Quantum Enhanced Simulated Annealing (QESA), a novel hybrid optimization framework that integrates quantum annealing (QA) into simulated annealing (SA) to tackle continuous optimization problems. While QA has shown promise in solving binary problems such as those expressed in Ising or QUBO form, its direct applicability to real-valued domains remains limited. QESA bridges this gap by using QA to select discrete search directions that guide SA through the continuous solution space, enabling the use of quantum resources without requiring full problem discretization. We demonstrate QESA's effectiveness on box-constrained quadratic programming (QP) problems, a class of non-convex optimization tasks that frequently arise in practice. Experimental results show that QESA consistently outperforms classical baselines in solution quality, particularly on larger and more ill-conditioned problems, while maintaining competitive runtime. As quantum annealing hardware matures, QESA offers a scalable and flexible strategy for leveraging quantum capabilities in continuous optimization.

Figures (5)

• Figure 1: Distribution of solution values for QESA solutions, grouped by diagonal scale. Each subplot corresponds to a different diagonal scaling factor applied to the matrix $Q$. The y-axis uses a logarithmic scale to highlight both dominant and rare value frequencies.
• Figure 2: Relative energy error for each method across different problem sizes and diagonal scales. Each subplot corresponds to a different diagonal scaling factor applied to the QP matrix. Energies are normalized by the corresponding Gurobi value.
• Figure 3: Average runtime per solver across different problem sizes (log scale on the y-axis). QESA_qpu reports cumulative quantum processing time, while all other methods reflect wall-clock time.
• Figure 4: Normalized final energy as a function of the number of QESA iterations, for problem sizes $n = 50, 100, 150$. Energies are normalized relative to the Gurobi solution. Each bar corresponds to a full QESA run using a temperature schedule matched to the given number of steps.
• Figure 5: Final QESA energy as a function of the probability $p$ of retaining each component of the QA-computed direction vector. Results are grouped by diagonal magnitude of the Q matrix. Lower energy indicates better performance.