Entanglement-assisted variational algorithm for discrete optimization problems

TL;DR

This work presents a quantum-inspired variational algorithm for discrete optimization, specifically targeting QUBO problems, by employing Generalized Group-Theoretic Coherent States to encode the state and allow analytic energy and gradient computations. The method captures nontrivial entanglement through a two-body coupling matrix while maintaining polynomial-time evaluability, enabling optimization of thousands of spins without Monte Carlo sampling. Benchmarking on 3D Edwards-Anderson instances shows competitive performance against state-of-the-art classical heuristics, with favorable scaling behavior as problem size grows. The results suggest that entanglement-aware variational Ansätze can provide scalable, effective alternatives for large-scale combinatorial optimization and may complement traditional heuristics in practical applications.

Abstract

From fundamental sciences to economics and industry, discrete optimization problems are ubiquitous. Yet, their complexity often renders exact solutions intractable, necessitating the use of approximate methods. Heuristics inspired by classical physics have long played a central role in this domain. More recently, quantum annealing has emerged as a promising alternative, with hardware implementations realized on both analog and digital quantum devices. Here, we develop a heuristic inspired by quantum annealing, using Generalized Coherent States as a parameterized variational Ansatz to represent the quantum state. This framework allows for the analytical computation of energy and gradients with low-degree polynomial complexity, enabling the study of large problems with thousands of spins. Concurrently, these states capture non-trivial entanglement, crucial for the effectiveness of quantum annealing. We benchmark the heuristic on the three-dimensional Edwards-Anderson model and compare the solution quality and runtime of our method to other popular heuristics. Our findings suggest that it offers a scalable way to leverage quantum effects for complex optimization problems, with the potential to complement or improve upon conventional alternatives in large-scale applications.

Figures (5)

• Figure 1: Sketch of the optimization scheme. As the Hamiltonian evolves, the parameters of the Ansatz are updated to minimize the expectation value of the Hamiltonian. Each time step, only one single update is performed.
• Figure 2: Time per iteration as a function of the non-zero elements in the adjacency matrix $W$. The number of non-zero elements is varied according to $N_{\mathrm{nnz}} = \qty(1 + \alpha\qty(N-1))N$. As the number of non-zero elements is decreased, the execution time per iteration crosses over from $\order{N^3}$ to $\order{N^2}$ scaling. The two dotted lines represent guides to the eye for the two scaling regimes. The simulations have been parallelized over $40$ threads.
• Figure 3: Median relative error $\varepsilon$ as a function of the number of spins $N$. Each method is run for $N_{\mathrm{t}} = 1000$ iterations on $1000$ random problem instances. The shaded regions represent the interquartile ranges of the resulting distributions.
• Figure 4: Median relative error $\varepsilon$ as a function of the number of iterations $N_{\mathrm{t}}$. Each method is benchmarked on $1000$ random problems. The shaded regions represent the interquartile range of the resulting distributions. We select a target error threshold (identified by the dashed line) and analyze the number of iterations required by GCS and PT-ICM to reach it (marked by the black dots).
• Figure 5: Resource analysis to reach a given error threshold.a) -- Number of iterations required by GCS and PT-ICM to reach an error threshold of $1.7\%$ as a function of the system size. b) -- Total runtime required by GCS and PT-ICM to reach an error threshold of $1.7\%$ as a function of the system size. Dashed lines indicate power-law fits to the data, with the corresponding scaling exponents annotated next to each line. GCS simulations have been parallelized over $40$ threads on a single computer.