Warm-starting quantum optimization

Daniel J. Egger1, Jakub Mareček2, and Stefan Woerner1

1IBM Quantum, IBM Research – Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
2Czech Technical University, Karlovo nam. 13, Prague 2, the Czech Republic

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Abstract

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

Many optimization problems in binary decision variables are hard to solve. In this work, we demonstrate how to leverage decades of research in classical optimization algorithms to warm-start quantum optimization algorithms. This allows the quantum algorithm to inherit the performance guarantees from the classical algorithm used in the warm-start.

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[2] Nathan Earnest, Caroline Tornow, and Daniel J. Egger, "Pulse-efficient circuit transpilation for quantum applications on cross-resonance-based hardware", Physical Review Research 3 4, 043088 (2021).

[3] Stefan H. Sack and Maksym Serbyn, "Quantum annealing initialization of the quantum approximate optimization algorithm", arXiv:2101.05742, Quantum 5, 491 (2021).

[4] Manuela Weigold, Johanna Barzen, Frank Leymann, and Daniel Vietz, Communications in Computer and Information Science 1429, 34 (2021) ISBN:978-3-030-87567-1.

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[6] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang, "Hybrid quantum-classical algorithms for approximate graph coloring", arXiv:2011.13420.

[7] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", arXiv:1912.04088.

[8] Amir M Aghaei, Bela Bauer, Kirill Shtengel, and Ryan V. Mishmash, "Efficient matrix-product-state preparation of highly entangled trial states: Weak Mott insulators on the triangular lattice revisited", arXiv:2009.12435.

[9] Zain H. Saleem, Teague Tomesh, Bilal Tariq, and Martin Suchara, "Approaches to Constrained Quantum Approximate Optimization", arXiv:2010.06660.

[10] Reuben Tate, Majid Farhadi, Creston Herold, Greg Mohler, and Swati Gupta, "Bridging Classical and Quantum with SDP initialized warm-starts for QAOA", arXiv:2010.14021.

[11] Jonathan Wurtz and Peter Love, "Classically optimal variational quantum algorithms", arXiv:2103.17065.

[12] Ioannis Kolotouros and Petros Wallden, "An evolving objective function for improved variational quantum optimisation", arXiv:2105.11766.

[13] Constantin Dalyac, Loïc Henriet, Emmanuel Jeandel, Wolfgang Lechner, Simon Perdrix, Marc Porcheron, and Margarita Veshchezerova, "Qualifying quantum approaches for hard industrial optimization problems. A case study in the field of smart-charging of electric vehicles", arXiv:2012.14859.

[14] M. Werninghaus, D. J. Egger, and S. Filipp, "High-Speed Calibration and Characterization of Superconducting Quantum Processors without Qubit Reset", PRX Quantum 2 2, 020324 (2021).

[15] Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, and Mohan Sarovar, "Lyapunov control-inspired strategies for quantum combinatorial optimization", arXiv:2108.05945.

[16] Sami Boulebnane, "Improving the Quantum Approximate Optimization Algorithm with postselection", arXiv:2011.05425.

[17] Johanna Barzen, "From Digital Humanities to Quantum Humanities: Potentials and Applications", arXiv:2103.11825.

[18] Daniel Beaulieu and Anh Pham, "Max-cut Clustering Utilizing Warm-Start QAOA and IBM Runtime", arXiv:2108.13464.

[19] L. C. G. Govia, C. Poole, M. Saffman, and H. K. Krovi, "Freedom of mixer rotation-axis improves performance in the quantum approximate optimization algorithm", arXiv:2107.13129.

[20] Stuart M. Harwood, Dimitar Trenev, Spencer T. Stober, Panagiotis Barkoutsos, Tanvi P. Gujarati, Sarah Mostame, and Donny Greenberg, "Improving the variational quantum eigensolver using variational adiabatic quantum computing", arXiv:2102.02875.

[21] Wim van Dam, Karim Eldefrawy, Nicholas Genise, and Natalie Parham, "Quantum Optimization Heuristics with an Application to Knapsack Problems", arXiv:2108.08805.

[22] Bryce Fuller, Charles Hadfield, Jennifer R. Glick, Takashi Imamichi, Toshinari Itoko, Richard J. Thompson, Yang Jiao, Marna M. Kagele, Adriana W. Blom-Schieber, Rudy Raymond, and Antonio Mezzacapo, "Approximate Solutions of Combinatorial Problems via Quantum Relaxations", arXiv:2111.03167.

The above citations are from Crossref's cited-by service (last updated successfully 2021-12-08 10:57:49) and SAO/NASA ADS (last updated successfully 2021-12-08 10:57:50). The list may be incomplete as not all publishers provide suitable and complete citation data.