Faster quantum and classical SDP approximations for quadratic binary optimization

Fernando G.S L. Brandão1,2,3, Richard Kueng1,2,4, and Daniel Stilck França5,6

1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
2Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA
3AWS Center for Quantum Computing, Pasadena, CA, USA
4Institute for Integrated Circuits, Johannes Kepler University Linz, Austria
5QMATH, Department of Mathematical Sciences, University of Copenhagen, Denmark
6Department of Mathematics, Technische Universität München, Germany

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Abstract

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.

Quadratic unconstrained binary optimization problems (QUBO) are prototypical NP-complete problems. Many hard combinatorial problems like the maximum cut of a graph can be easily formulated in this framework, and they find applications in many areas of science and industry. We don’t expect computers – including quantum computers – to solve general QUBOs efficiently. However, the pioneering work of Goemans and Williamson has shown that they do admit an efficient relaxation in the form of a semidefinite program (SDP). Remarkably, by solving this relaxation, it is possible to obtain a solution that is a constant fraction away from the optimal one.

In contrast to QUBOs, classical SDP solvers do not require exponential resources. Runtime and memory do, however, scale superlinearly in problem size (dimension and number of constraints). Recent works have shown that quantum computers hold the promise of further speeding up the solution of SDPs. However, these assertions depend on the underlying problem structure. When applied to SDP relaxations of QUBOs, the obtained runtime is not competitive.

This work bridges this gap and shows how to obtain quantum speedups for this important class of problems by developing a new quantum algorithm. It has a better runtime for random instance of QUBOs when compared to both existing classical and quantum approaches. To achieve this speedup, the authors exploit that the constraints of the SDP have a natural interpretation in quantum information theory: the feasible points are approximately indistinguishable from the maximally mixed state when measured in the computational basis. This interpretation gives rise to a simple way of checking whether a candidate solution is feasible on a quantum computer and how to update it accordingly if it is not.

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[3] Jonathan Wei Zhong Lau, Kian Hwee Lim, Kishor Bharti, Leong-Chuan Kwek, and Sai Vinjanampathy, "Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor", Physical Review Letters 130 24, 240601 (2023).

[4] Álvaro M. Alhambra, "Quantum Many-Body Systems in Thermal Equilibrium", PRX Quantum 4 4, 040201 (2023).

[5] Alexander M. Dalzell, B. David Clader, Grant Salton, Mario Berta, Cedric Yen-Yu Lin, David A. Bader, Nikitas Stamatopoulos, Martin J. A. Schuetz, Fernando G. S. L. Brandão, Helmut G. Katzgraber, and William J. Zeng, "End-To-End Resource Analysis for Quantum Interior-Point Methods and Portfolio Optimization", PRX Quantum 4 4, 040325 (2023).

[6] Taylor L. Patti, Jean Kossaifi, Anima Anandkumar, and Susanne F. Yelin, "Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints", Quantum 7, 1057 (2023).

[7] Rodolfo A. Quintero and Luis F. Zuluaga, Encyclopedia of Optimization 1 (2023) ISBN:978-3-030-54621-2.

[8] Brandon Augustino, Giacomo Nannicini, Tamás Terlaky, and Luis F. Zuluaga, "Quantum Interior Point Methods for Semidefinite Optimization", Quantum 7, 1110 (2023).

[9] Daniel Stilck França and Raul García-Patrón, "Limitations of optimization algorithms on noisy quantum devices", Nature Physics 17 11, 1221 (2021).

[10] Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara, and Mehdi Soleimanifar, "Sample-efficient learning of interacting quantum systems", Nature Physics 17 8, 931 (2021).

[11] Daniel J. Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Simonetto, Stefan Woerner, and Elena Yndurain, "Quantum Computing for Finance: State of the Art and Future Prospects", arXiv:2006.14510, (2020).

[12] Arbel Haim, Richard Kueng, and Gil Refael, "Variational-Correlations Approach to Quantum Many-body Problems", arXiv:2001.06510, (2020).

[13] Simon Apers and Ronald de Wolf, "Quantum Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving", arXiv:1911.07306, (2019).

[14] Claudio Gambella and Andrea Simonetto, "Multi-block ADMM Heuristics for Mixed-Binary Optimization on Classical and Quantum Computers", arXiv:2001.02069, (2020).

[15] Fernando G. S. L. Brandão, Richard Kueng, and Daniel Stilck França, "Fast and robust quantum state tomography from few basis measurements", arXiv:2009.08216, (2020).

[16] Iordanis Kerenidis, Anupam Prakash, and Dániel Szilágyi, "Quantum algorithms for Second-Order Cone Programming and Support Vector Machines", arXiv:1908.06720, (2019).

[17] Iordanis Kerenidis, Anupam Prakash, and Dániel Szilágyi, "Quantum algorithms for Second-Order Cone Programming and Support Vector Machines", Quantum 5, 427 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-29 07:15:40) and SAO/NASA ADS (last updated successfully 2024-03-29 07:15:41). The list may be incomplete as not all publishers provide suitable and complete citation data.