[Paper Review] Comparing the Digital Annealer with Classical Evolutionary Algorithm

Mayowa Ayodele|arXiv (Cornell University)|May 26, 2022

Evolutionary Algorithms and Applications4 citations

TL;DR

This paper compares Fujitsu's Digital Annealer (DA), a quantum-inspired hardware solver, against a classical Genetic Algorithm (GA) on combinatorial optimization problems like the Travelling Salesman Problem (TSP), Quadratic Assignment Problem (QAP), and Multi-dimensional Knapsack Problem (MKP). Despite GA's fast implementation, the DA achieves significantly better average objective function values, especially on QAP instances, with a speedup exceeding 10x, demonstrating the advantage of specialized hardware for certain problem types.

ABSTRACT

In more recent years, there has been increasing research interest in exploiting the use of application specific hardware for solving optimisation problems. Examples of solvers that use specialised hardware are IBM's Quantum System One and D-wave's Quantum Annealer (QA) and Fujitsu's Digital Annealer (DA). These solvers have been developed to optimise problems faster than traditional meta-heuristics implemented on general purpose machines. Previous research has shown that these solvers (can optimise many problems much quicker than exact solvers such as GUROBI and CPLEX. Such conclusions have not been made when comparing hardware solvers with classical evolutionary algorithms. Making a fair comparison between traditional evolutionary algorithms, such as Genetic Algorithm (GA), and the DA (or other similar solvers) is challenging because the later benefits from the use of application specific hardware while evolutionary algorithms are often implemented on general-purpose machines. Moreover, quantum or quantum-inspired solvers are limited to solving problems in a specific format. A common formulation used is Quadratic Unconstrained Binary Optimisation (QUBO). Many optimisation problems are however constrained and have natural representations that are non-binary. Converting such problems to QUBO can lead to more problem difficulty and/or larger search space. The question addressed in this paper is whether quantum or quantum-inspired solvers can optimise QUBO transformations of combinatorial optimisation problems faster than classical evolutionary algorithms applied to the same problems in their natural representations. We show that the DA often present better average objective function values than GA on Travelling Salesman, Quadratic Assignment and Multi-dimensional Knapsack Problem instances.

Motivation & Objective

• To evaluate whether quantum-inspired hardware solvers like the Digital Annealer (DA) outperform classical evolutionary algorithms such as Genetic Algorithms (GA) on combinatorial optimization problems.
• To conduct a fair comparison between DA and GA by using the same problem instances and time limits, despite differences in hardware architecture and implementation.
• To investigate whether the natural representation of problems (e.g., permutation or binary) affects the relative performance of DA versus GA.
• To assess the impact of problem formulation—particularly QUBO transformation—on the performance gap between hardware-optimized solvers and classical meta-heuristics.

Proposed method

• The study uses the Digital Annealer (DA), a CMOS-based quantum-inspired hardware solver, to solve Quadratic Unconstrained Binary Optimization (QUBO) formulations of TSP, QAP, and MKP.
• A standard Genetic Algorithm (GA) is implemented on a high-performance computer, using standard operators: selection, crossover, and mutation, with problem-specific representations such as random keys and permutation matrices.
• Problems are converted to QUBO format using two-way one-hot encoding for permutation problems and standard binary encoding for knapsack problems.
• Optuna is used to hyperparameter-tune both DA and GA for each instance and time limit (1s, 2s, 5s, 10s), ensuring optimal configuration for fair comparison.
• Performance is evaluated across 20 independent runs per instance, measuring average fitness and standard deviation, with optimal values marked in bold.
• The comparison is conducted across three problem types: TSP (permutation), QAP (permutation), and MKP (constrained binary), all transformed into QUBO.

Experimental results

Research questions

• RQ1 Does the Digital Annealer (DA) achieve better solution quality than a classical Genetic Algorithm (GA) on QUBO-transformed combinatorial optimization problems?
• RQ2 How does the performance of DA compare to GA across different problem types (TSP, QAP, MKP) and time limits?
• RQ3 Can classical evolutionary algorithms like GA outperform DA in specific cases, especially when given more time to converge?
• RQ4 To what extent does the QUBO transformation affect the performance gap between hardware-optimized solvers and classical meta-heuristics?

Key findings

• On QAP instances, the DA achieved a speedup of more than 10x over the GA, with significantly better average fitness values across all time limits.
• For TSP, the DA reached optimal solutions on 6 out of 10 instances within 1 second, while the GA achieved optimal on only 1 instance; DA consistently outperformed GA in average fitness at 1s and 2s.
• At 10 seconds, the DA achieved better or equal average fitness than GA on 7 out of 10 TSP instances, with GA outperforming DA on only 2 instances.
• On MKP instances, the DA showed consistent but less pronounced performance gains over GA, indicating that the problem structure plays a key role in hardware advantage.
• The DA's performance was particularly strong on QAP due to its naturally quadratic structure, which aligns well with QUBO formulations.
• Despite fast implementation, GA improved faster on some TSP instances (e.g., berlin52, brazil58), showing that convergence speed can vary significantly by problem instance.