Quantum optimisation applied to the Quadratic Assignment Problem

TL;DR

This work evaluates the non-variational Quantum Walk-based Optimisation Algorithm (NV-QWOA) for solving small instances of the Quadratic Assignment Problem (QAP) and benchmarks it against classical heuristics (Max-Min Ant System and Greedy Local Search) and a Grover-based quantum baseline. NV-QWOA leverages a continuous-time quantum walk mixer with a phase-shifting cost unitary, using only three hyperparameters and a transposition-based mixing graph to amplify optimal and near-optimal solutions, while avoiding heavy parameter tuning typical of variational approaches. Key findings show NV-QWOA achieving favorable scaling in the number of objective-function evaluations and maintaining a consistent 10% optimal-solution probability across problem sizes $n=5$–$10$, with the required circuit depth $p$ growing roughly as $p\approx 0.0049\,n^{4}$. The results also reveal linear growth in internode distance and strong amplification of optimal shells, supporting the practical utility of quantum walks for combinatorial optimization on small instances, while highlighting the need for noise-aware hardware studies and scalable parameter-transfer strategies for larger problems.

Abstract

This paper investigates the performance of the emerging non-variational Quantum Walk-based Optimisation Algorithm (NV-QWOA) for solving small instances of the Quadratic Assignment Problem (QAP). NV-QWOA is benchmarked against classical heuristics, the MaxMin Ant System (MMAS) and Greedy Local Search (GLS), as well as the Grover quantum search algorithm, which serves as a quantum baseline. Performance is evaluated using two metrics: the number of objective function evaluations and the number of algorithm iterations required to consistently reach optimal or near optimal solutions across QAP instances with 5 to 10 facilities. The motivation for this study stems from limitations of both classical exact methods and current quantum algorithms. Variational Quantum Algorithms (VQAs), such as QAOA and VQE, while widely studied, suffer from costly parameter tuning and barren plateaus that hinder convergence. By adopting a non-variational approach, this work explores a potentially more efficient and scalable quantum strategy for combinatorial optimisation. The results provide a direct comparative analysis between classical and quantum frameworks, characterising the average case performance of NV-QWOA. Our findings highlight the practical utility of quantum walks for complex combinatorial problems and establish a foundation for future quantum optimisation algorithms.

Figures (8)

• Figure 1: Facility flows and location distances of an example $n = 4$ QAP
• Figure 2: The transposition graph for $n = 4$ QAP instances, centered at solution $0 = [0,1,2,3]$. The nodes are the solutions, and the edges connect nearest-neighbour solutions i.e those that differ by one transposition.
• Figure 3: The mean number of objective function evaluations for the MMAS algorithm over problem sizes $5,6,7,8,9,10$
• Figure 4: Direct comparison of GLS with NV-QWOA. Left: The solve and measurement probability for the GLS and NV-QWOA respectively over problem sizes $5,6,7,8,9,10$. Right: The mean number of objective function evaluations for the GLS and NV-QWOA over problem sizes $5,6,7,8,9,10$
• Figure 5: Optimal solution probability violin plots across various $p$ value ranges