NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization

M. R. Perelshtein1,2,3, A. I. Pakhomchik1, Ar. A. Melnikov1, M. Podobrii1, A. Termanova1, I. Kreidich1, B. Nuriev1, S. Iudin1, C. W. Mansell1, and V. M. Vinokur1,4

1Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland
2QTF Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland
3InstituteQ – the Finnish Quantum Institute, Aalto University, Finland
4Physics Department, City College of the City University of New York, 160 Convent Ave, New York, NY 10031, USA

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Quantum algorithms are getting extremely popular due to their potential to significantly outperform classical algorithms. Yet, applying quantum algorithms to optimization problems meets challenges related to the efficiency of quantum algorithms training, the shape of their cost landscape, the accuracy of their output, and their ability to scale to large-size problems. Here, we present an approximate gradient-based quantum algorithm for hardware-efficient circuits with amplitude encoding. We show how simple linear constraints can be directly incorporated into the circuit without additional modification of the objective function with penalty terms. We employ numerical simulations to test it on $\texttt{MaxCut}$ problems with complete weighted graphs with thousands of nodes and run the algorithm on a superconducting quantum processor. We find that for unconstrained $\texttt{MaxCut}$ problems with more than 1000 nodes, the hybrid approach combining our algorithm with a classical solver called CPLEX can find a better solution than CPLEX alone. This demonstrates that hybrid optimization is one of the leading use cases for modern quantum devices.

Optimization is the process of adjusting systems and operations to make them more efficient and effective. Imagine, for example, a control panel in a factory with lots of settings. Finding out how to adjust the settings to make the factory as energy efficient as possible would constitute an optimization task. Developing better optimization algorithms, both classical and quantum, is an important area of research.

It is often useful to imagine each combination of settings as corresponding to a position on a map. The quantity being optimized — the energy efficiency in the previous example — would be represented by the height above sea level of the different map positions. In prior work, an efficient way of encoding optimization problems into quantum processors was combined with a gradient-based method (i.e., a method that uses the steepness or shallowness of the terrain to decide the next settings to try).

We build on this prior work by incorporating simple linear constraints into the problem. This is useful because it is usually the case that not every combination of settings is physically possible. Hence, the available options need to be constrained. Importantly, as shown by the analysis in the paper, our way of providing the constraints does not make the optimization problem more difficult or complicated.

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Cited by

[1] Ar A Melnikov, A A Termanova, S V Dolgov, F Neukart, and M R Perelshtein, "Quantum state preparation using tensor networks", Quantum Science and Technology 8 3, 035027 (2023).

[2] Atsushi Matsuo, Yudai Suzuki, Ikko Hamamura, and Shigeru Yamashita, "Enhancing VQE Convergence for Optimization Problems with Problem-Specific Parameterized Quantum Circuits", IEICE Transactions on Information and Systems 106 11, 1772 (2023).

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