Continuous-time quantum walks for MAX-CUT are hot

Robert J. Banks1, Ehsan Haque2, Farah Nazef2, Fatima Fethallah2, Fatima Ruqaya2, Hamza Ahsan2, Het Vora2, Hibah Tahir2, Ibrahim Ahmad2, Isaac Hewins2, Ishaq Shah2, Krish Baranwal2, Mannan Arora2, Mateen Asad2, Mubasshirah Khan2, Nabian Hasan2, Nuh Azad2, Salgai Fedaiee2, Shakeel Majeed2, Shayam Bhuyan2, Tasfia Tarannum2, Yahya Ali2, Dan E. Browne3, and P. A. Warburton1,4

1London Centre for Nanotechnology, UCL, London WC1H 0AH, UK
2Newham Collegiate Sixth Form Centre, 326 Barking Rd, London, E6 2BB, UK
3Department of Physics and Astronomy, UCL, London WC1E 6BT, UK
4Department of Electronic & Electrical Engineering, UCL, London WC1E 7JE, UK

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By exploiting the link between time-independent Hamiltonians and thermalisation, heuristic predictions on the performance of continuous-time quantum walks for MAX-CUT are made. The resulting predictions depend on the number of triangles in the underlying MAX-CUT graph. We extend these results to the time-dependent setting with multi-stage quantum walks and Floquet systems. The approach followed here provides a novel way of understanding the role of unitary dynamics in tackling combinatorial optimisation problems with continuous-time quantum algorithms.

Combinatorial optimisation problems feature in many aspects of modern-day life. Examples include finding the shortest path, maximizing profit and optimally scheduling deliveries. These problems are typically difficult to solve. Here we focus on the canonical problem known as MAX-CUT. Continuous-time quantum walks present a novel way of tackling optimisation problems by exploiting quantum effects. In this paper we discuss how to optimise continuous-time quantum walks for MAX-CUT.

Continuous-time quantum walks contain a free parameter. A well-optimised parameter results in a better quality of solution. In order to optimise the quantum walk, we utilise the well-established hypothesis that closed systems can thermalise. The associated temperature turns out to be high. By effectively modelling the density of states for the quantum walk we can reliably estimate the optimal choice of free parameter without a (classical) variational outer-loop. Importantly, the estimated optimal choice of the free parameter can be tied to properties of the underlying MAX-CUT graph.

This work presents a novel approach, combining statistical physics with quantum optimization. Future work might involve extending the insights in this paper to a broader range of quantum approaches to optimisation.

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[1] Sebastian Schulz, Dennis Willsch, and Kristel Michielsen, "Guided quantum walk", Physical Review Research 6 1, 013312 (2024).

[2] R. Au-Yeung, B. Camino, O. Rathore, and V. Kendon, "Quantum algorithms for scientific applications", arXiv:2312.14904, (2023).

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