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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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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