Applying quantum algorithms to constraint satisfaction problems

Earl Campbell1, Ankur Khurana2,3, and Ashley Montanaro4

1Department of Physics and Astronomy, University of Sheffield, Sheffield, UK
2Quantum Engineering Centre for Doctoral Training, University of Bristol, UK
3School of Physics, University of Bristol, UK
4School of Mathematics, University of Bristol, UK

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Abstract

Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best classical algorithms and taking into account realistic hardware parameters and overheads for fault-tolerance. All known examples of such speedups correspond to problems related to simulation of quantum systems and cryptography. Here we apply general-purpose quantum algorithms for solving constraint satisfaction problems to two families of prototypical NP-complete problems: boolean satisfiability and graph colouring. We consider two quantum approaches: Grover's algorithm and a quantum algorithm for accelerating backtracking algorithms. We compare the performance of optimised versions of these algorithms, when applied to random problem instances, against leading classical algorithms. Even when considering only problem instances that can be solved within one day, we find that there are potentially large quantum speedups available. In the most optimistic parameter regime we consider, this could be a factor of over $10^5$ relative to a classical desktop computer; in the least optimistic regime, the speedup is reduced to a factor of over $10^3$. However, the number of physical qubits used is extremely large, and improved fault-tolerance methods will likely be needed to make these results practical. In particular, the quantum advantage disappears if one includes the cost of the classical processing power required to perform decoding of the surface code using current techniques.

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Cited by

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[2] Simon Martiel and Maxime Remaud, Lecture Notes in Computer Science 12011, 597 (2020) ISBN:978-3-030-38918-5.

[3] J. Haferkamp, D. Hangleiter, A. Bouland, B. Fefferman, J. Eisert, and J. Bermejo-Vega, "Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dynamics", Physical Review Letters 125 25, 250501 (2020).

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[5] Rhea Parekh, Andrea Ricciardi, Ahmed Darwish, and Stephen DiAdamo, 2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS) 9 (2021) ISBN:978-1-7281-8674-0.

[6] Karuna Kadian, Sunita Garhwal, and Ajay Kumar, "Quantum walk and its application domains: A systematic review", Computer Science Review 41, 100419 (2021).

[7] Yan Liang and Bingo Wing-Kuen Ling, "Infinite Impulse Response Filter Bank Based Graphic Equalizer Design via Functional Inequality Constrained Optimization and Genetic Algorithm", IEEE Access 9, 65116 (2021).

[8] Iñaki Fernández Pérez, Fernando de la Prieta, Sara Rodríguez-González, Juan M. Corchado, and Javier Prieto, Lecture Notes in Networks and Systems 603, 155 (2023) ISBN:978-3-031-22355-6.

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[13] Kyle E. C. Booth, Bryan O'Gorman, Jeffrey Marshall, Stuart Hadfield, and Eleanor Rieffel, "Quantum-accelerated constraint programming", Quantum 5, 550 (2021).

[14] Mathys Rennela, Sebastiaan Brand, Alfons Laarman, and Vedran Dunjko, "Hybrid divide-and-conquer approach for tree search algorithms", Quantum 7, 959 (2023).

[15] Kyle E. C. Booth, Bryan O’Gorman, Jeffrey Marshall, Stuart Hadfield, and Eleanor Rieffel, Lecture Notes in Computer Science 12333, 72 (2020) ISBN:978-3-030-58474-0.

[16] Yuval R. Sanders, Dominic W. Berry, Pedro C.S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush, "Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization", PRX Quantum 1 2, 020312 (2020).

[17] Ashley Montanaro, "Quantum speedup of branch-and-bound algorithms", Physical Review Research 2 1, 013056 (2020).

[18] James R. Wootton, 2020 IEEE Conference on Games (CoG) 73 (2020) ISBN:978-1-7281-4533-4.

[19] Cezar-Mihail Alexandru, Ella Bridgett-Tomkinson, Noah Linden, Joseph MacManus, Ashley Montanaro, and Hannah Morris, "Quantum speedups of some general-purpose numerical optimisation algorithms", Quantum Science and Technology 5 4, 045014 (2020).

[20] Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis, "Optimizing Quantum Error Correction Codes with Reinforcement Learning", Quantum 3, 215 (2019).

[21] Yimin Ge and Vedran Dunjko, "A hybrid algorithm framework for small quantum computers with application to finding Hamiltonian cycles", Journal of Mathematical Physics 61 1, 012201 (2020).

[22] Shraddha Singh, Andrew S. Darmawan, Benjamin J. Brown, and Shruti Puri, "High-fidelity magic-state preparation with a biased-noise architecture", Physical Review A 105 5, 052410 (2022).

[23] Earl T Campbell, "Early fault-tolerant simulations of the Hubbard model", Quantum Science and Technology 7 1, 015007 (2022).

[24] Furqan Ahmed and Petri Mahonen, 2021 IEEE 32nd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC) 1128 (2021) ISBN:978-1-7281-7586-7.

[25] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", PRX Quantum 3 1, 010329 (2022).

[26] Christoph Roch, Santiago Londono Castillo, and Claudia Linnhoff-Popien, 2022 IEEE 19th International Conference on Software Architecture Companion (ICSA-C) 147 (2022) ISBN:978-1-6654-9493-9.

[27] Craig Gidney and Martin Ekerå, "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", Quantum 5, 433 (2021).

[28] Jarrod R. McClean, Matthew P. Harrigan, Masoud Mohseni, Nicholas C. Rubin, Zhang Jiang, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, "Low-Depth Mechanisms for Quantum Optimization", PRX Quantum 2 3, 030312 (2021).

[29] Lawrence Z. Cohen, Isaac H. Kim, Stephen D. Bartlett, and Benjamin J. Brown, "Low-overhead fault-tolerant quantum computing using long-range connectivity", Science Advances 8 20, eabn1717 (2022).

[30] E. J. Crosson and D. A. Lidar, "Prospects for quantum enhancement with diabatic quantum annealing", Nature Reviews Physics 3 7, 466 (2021).

[31] Alexander M. Dalzell, Nicola Pancotti, Earl T. Campbell, and Fernando G.S.L. Brandão, Proceedings of the 55th Annual ACM Symposium on Theory of Computing 1131 (2023) ISBN:9781450399135.

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[33] Earl Campbell, "Random Compiler for Fast Hamiltonian Simulation", Physical Review Letters 123 7, 070503 (2019).

[34] Craig Gidney and Martin Ekerå, "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", arXiv:1905.09749, (2019).

[35] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G. S. L. Brandão, "Building a fault-tolerant quantum computer using concatenated cat codes", arXiv:2012.04108, (2020).

[36] Hendrik Poulsen Nautrup, Nicolas Delfosse, Vedran Dunjko, Hans J. Briegel, and Nicolai Friis, "Optimizing Quantum Error Correction Codes with Reinforcement Learning", arXiv:1812.08451, (2018).

[37] Craig Gidney, "Windowed quantum arithmetic", arXiv:1905.07682, (2019).

[38] Sami Boulebnane and Ashley Montanaro, "Solving boolean satisfiability problems with the quantum approximate optimization algorithm", arXiv:2208.06909, (2022).

[39] Chris Cade, Marten Folkertsma, Ido Niesen, and Jordi Weggemans, "Quantifying Grover speed-ups beyond asymptotic analysis", arXiv:2203.04975, (2022).

[40] Sukin Sim, Yudong Cao, Jonathan Romero, Peter D. Johnson, and Alan Aspuru-Guzik, "A framework for algorithm deployment on cloud-based quantum computers", arXiv:1810.10576, (2018).

[41] Alexander M. Dalzell, Nicola Pancotti, Earl T. Campbell, and Fernando G. S. L. Brandão, "Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end", arXiv:2212.01513, (2022).

[42] Charles Yuan, Agnes Villanyi, and Michael Carbin, "Quantum Control Machine: The Limits of Quantum Programs as Data", arXiv:2304.15000, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-06-09 04:56:04) and SAO/NASA ADS (last updated successfully 2023-06-09 04:56:05). The list may be incomplete as not all publishers provide suitable and complete citation data.