Variational Quantum Algorithms for Semidefinite Programming

Dhrumil Patel; Patrick J. Coles; Mark M. Wilde

doi:10.22331/q-2024-06-17-1374

Variational Quantum Algorithms for Semidefinite Programming

Dhrumil Patel¹, Patrick J. Coles², and Mark M. Wilde^3,4

¹Department of Computer Science, Cornell University, Ithaca, New York, 14850, USA
²Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
³Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
⁴School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA

Published:	2024-06-17, volume 8, page 1374
Eprint:	arXiv:2112.08859v3
Doi:	https://doi.org/10.22331/q-2024-06-17-1374
Citation:	Quantum 8, 1374 (2024).

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Abstract

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., $N \gg M$, where $N$ is the dimension of the input matrices and $M$ is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.

Featured image: Depiction of the various reductions employed in our variational quantum algorithm for semi-definite programming. The three trace terms in the final objective function at the bottom can be evaluated by means of measuring observables on the states generated by parameterized quantum circuits.

Popular summary

Many real-world problems in science and industry can be expressed as optimization problems, which involve finding the best solution while meeting specific constraints. Among these, a special class of optimization problems called semidefinite programming holds significance. They are widely used to model or approximate problems arising in various fields such as operations research, combinatorial optimization, control theory, and quantum information theory. For solving these programs, quantum algorithms have been proven to provide a quadratic speedup over classical algorithms. However, these quantum algorithms are not well-suited for current quantum devices, which are noisy and limited in their capabilities. In this work, we propose three quantum algorithms designed to run on these noisy devices. Our algorithms are hybrid quantum-classical algorithms that have a classical computer available for optimization, only calling a quantum computer for tasks that are not efficiently solvable by it. We rigorously analyze the performance of one of our algorithms, quantifying how rapidly it converges to the optimal value. Finally, to demonstrate their practicality, we numerically simulate our quantum algorithms for problems like MaxCut, a prominent graph theoretic problem. Our simulations showcase their effectiveness even in the presence of noise.

► BibTeX data

@article{Patel2024variationalquantum,
  doi = {10.22331/q-2024-06-17-1374},
  url = {https://doi.org/10.22331/q-2024-06-17-1374},
  title = {Variational {Q}uantum {A}lgorithms for {S}emidefinite {P}rogramming},
  author = {Patel, Dhrumil and Coles, Patrick J. and Wilde, Mark M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1374},
  month = jun,
  year = {2024}
}

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