Entanglement-assisted variational algorithm for discrete optimization problems (2501.09078v1)

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, potentially surpassing conventional alternatives in large-scale applications.

• The paper introduces a quantum-inspired heuristic for discrete optimization problems like QUBO, utilizing Generalized Coherent States as a variational Ansatz inspired by quantum annealing.
• The proposed method efficiently incorporates entanglement and allows for analytical computation of energy and gradients, resulting in polynomial computational complexity.
• Numerical results on the 3D Edwards-Anderson model show the heuristic often surpasses conventional methods like Simulated Annealing and Local Quantum Annealing, particularly for larger problem sizes.

Overview of "Entanglement-assisted variational algorithm for discrete optimization problems"

The paper, authored by Lorenzo Fioroni and Vincenzo Savona, presents a novel approach to tackle discrete optimization problems, specifically focusing on the challenging class of Quadratic Unconstrained Binary Optimization (QUBO) problems, which are classified as NP-Hard. The paper is grounded in the field of quantum computing, presenting a heuristic inspired by quantum annealing and utilizing Generalized Coherent States (GCS) as a variational Ansatz.

Quantum annealing has gained traction as a potential strategy to address hard optimization challenges by preparing the ground state of a quantum Hamiltonian over time adiabatically. This research leverages the coherent states framework to incorporate entanglement, a key resource for quantum computation, into the heuristic. The paper introduces a variational method emulating quantum annealing dynamics analytically and benchmarks it against existing algorithms such as Simulated Annealing (SA), Local Quantum Annealing (LQA), and Parallel Tempering with Iso-energetic Cluster Moves (PT-ICM) on the three-dimensional Edwards-Anderson model.

Technical Contributions

• Quantum-inspired Heuristic: The authors devised a heuristic inspired by quantum annealing, employing Generalized Coherent States (GCS) as a variational Ansatz. This approach permits efficient analytical computation of energy and gradients, facilitating the paper of optimization problems with up to thousands of variables.
• Entanglement Inclusion: By adopting GCS, which encompass product states and allow for entanglement, the methodology captures non-trivial quantum correlations. This approach aims to leverage potential quantum advantages for optimization tasks.
• Analytical and Scalable Evaluation: The proposed method computes the loss function and its gradients analytically, circumventing the need for resource-intensive Monte Carlo sampling. This results in a heuristic with a polynomial computational complexity that scales as $\mathcal{O}(N^3)$ for dense matrices and $\mathcal{O}(N^2)$ for sparse matrices like those in the studied Edwards-Anderson model.

Numerical Results

The heuristic was evaluated by comparing the solution quality and computational efficiency against established methods on random instances of the three-dimensional Edwards-Anderson model. The results showed that this Quantum-inspired approach often surpasses conventional methods such as SA and LQA in terms of the relative error in solution quality. Specifically, for larger problem sizes, the proposed heuristic outperforms standard approaches, though PT-ICM may yield better results in smaller instances.

Implications and Potential Developments

This research marks a significant step in bridging quantum physics principles with classical heuristic design for optimization problems. The ability to efficiently incorporate elements of quantum mechanics, such as entanglement, into classical computations opens new possibilities for complex problem-solving where traditional methods falter. A future direction suggested by the paper includes enhancing the Ansatz's expressivity or simplifying it for specific problem topologies to further refine computational efficiency and solution accuracy.

By expanding the applicability of quantum-inspired methodologies, this paper offers a promising outlook on the integration of quantum computational paradigms into large-scale, classical optimization scenarios. As quantum technologies continue to mature, the insights and frameworks from this research may contribute significantly to the development of more robust, efficient algorithms capable of addressing an even broader range of NP-Hard problems.