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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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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Cited by

[1] Nima Nikmehr, Peng Zhang, and Mikhail A. Bragin, "Quantum Distributed Unit Commitment: An Application in Microgrids", IEEE Transactions on Power Systems 37 5, 3592 (2022).

[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] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[4] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics, and Astronomy 66 5, 250302 (2023).

[5] Johannes Weidenfeller, Lucia C. Valor, Julien Gacon, Caroline Tornow, Luciano Bello, Stefan Woerner, and Daniel J. Egger, "Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware", Quantum 6, 870 (2022).

[6] Takuya Yoshioka, Keita Sasada, Yuichiro Nakano, and Keisuke Fujii, "Fermionic quantum approximate optimization algorithm", Physical Review Research 5 2, 023071 (2023).

[7] James Dborin, Fergus Barratt, Vinul Wimalaweera, Lewis Wright, and Andrew G. Green, "Matrix product state pre-training for quantum machine learning", Quantum Science and Technology 7 3, 035014 (2022).

[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, (2020).

[9] G. Wendin, "Quantum information processing with superconducting circuits: a perspective", arXiv:2302.04558, (2023).

[10] Phillip C. Lotshaw, Thien Nguyen, Anthony Santana, Alexander McCaskey, Rebekah Herrman, James Ostrowski, George Siopsis, and Travis S. Humble, "Scaling quantum approximate optimization on near-term hardware", Scientific Reports 12, 12388 (2022).

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

[12] 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, (2021).

[13] Samuel Duffield, Marcello Benedetti, and Matthias Rosenkranz, "Bayesian learning of parameterised quantum circuits", Machine Learning: Science and Technology 4 2, 025007 (2023).

[14] 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, (2020).

[15] Laurin E. Fischer, Daniel Miller, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Daniel J. Egger, and Ivano Tavernelli, "Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space", Physical Review Research 4 3, 033027 (2022).

[16] Jonathan Wurtz and Peter J. Love, "Counterdiabaticity and the quantum approximate optimization algorithm", Quantum 6, 635 (2022).

[17] Noah L. Wach, Manuel S. Rudolph, Fred Jendrzejewski, and Sebastian Schmitt, "Data re-uploading with a single qudit", arXiv:2302.13932, (2023).

[18] Christa Zoufal, Ryan V. Mishmash, Nitin Sharma, Niraj Kumar, Aashish Sheshadri, Amol Deshmukh, Noelle Ibrahim, Julien Gacon, and Stefan Woerner, "Variational quantum algorithm for unconstrained black box binary optimization: Application to feature selection", Quantum 7, 909 (2023).

[19] 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).

[20] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar, "Graph neural network initialisation of quantum approximate optimisation", Quantum 6, 861 (2022).

[21] Johanna Barzen, "From Digital Humanities to Quantum Humanities: Potentials and Applications", arXiv:2103.11825, (2021).

[22] Ioannis Kolotouros and Petros Wallden, "Evolving objective function for improved variational quantum optimization", Physical Review Research 4 2, 023225 (2022).

[23] Archismita Dalal and Amara Katabarwa, "Noise tailoring for robust amplitude estimation", New Journal of Physics 25 2, 023015 (2023).

[24] Teague Tomesh, Zain H. Saleem, and Martin Suchara, "Quantum Local Search with the Quantum Alternating Operator Ansatz", Quantum 6, 781 (2022).

[25] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", arXiv:1912.04088, (2019).

[26] Elijah Pelofske, "Mapping state transition susceptibility in quantum annealing", Physical Review Research 5 1, 013224 (2023).

[27] Zain H. Saleem, Teague Tomesh, Bilal Tariq, and Martin Suchara, "Approaches to Constrained Quantum Approximate Optimization", arXiv:2010.06660, (2020).

[28] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", Quantum 5, 428 (2021).

[29] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang, "Hybrid quantum-classical algorithms for approximate graph coloring", Quantum 6, 678 (2022).

[30] 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, (2020).

[31] Daniel Beaulieu and Anh Pham, "Max-cut Clustering Utilizing Warm-Start QAOA and IBM Runtime", arXiv:2108.13464, (2021).

[32] Wim van Dam, Karim Eldefrawy, Nicholas Genise, and Natalie Parham, "Quantum Optimization Heuristics with an Application to Knapsack Problems", arXiv:2108.08805, (2021).

[33] Nicolas PD Sawaya, Albert T Schmitz, and Stuart Hadfield, "Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems", arXiv:2203.14432, (2022).

[34] Franz G. Fuchs, Kjetil Olsen Lye, Halvor Møll Nilsen, Alexander J. Stasik, and Giorgio Sartor, "Constrained mixers for the quantum approximate optimization algorithm", arXiv:2203.06095, (2022).

[35] Jonathan Wurtz and Peter Love, "Classically optimal variational quantum algorithms", arXiv:2103.17065, (2021).

[36] Vicente P. Soloviev, Concha Bielza, and Pedro Larrañaga, "Quantum approximate optimization algorithm for Bayesian network structure learning", Quantum Information Processing 22 1, 19 (2023).

[37] 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, (2021).

[38] Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, and Mohan Sarovar, "Lyapunov-control-inspired strategies for quantum combinatorial optimization", Physical Review A 106 6, 062414 (2022).

[39] Sami Boulebnane, "Improving the Quantum Approximate Optimization Algorithm with postselection", arXiv:2011.05425, (2020).

[40] Daniel Beaulieu and Anh Pham, "Evaluating performance of hybrid quantum optimization algorithms for MAXCUT Clustering using IBM runtime environment", arXiv:2112.03199, (2021).

[41] Elijah Pelofske, Georg Hahn, and Hristo Djidjev, "Initial state encoding via reverse quantum annealing and h-gain features", arXiv:2303.13748, (2023).

[42] Lilly Palackal, Benedikt Poggel, Matthias Wulff, Hans Ehm, Jeanette Miriam Lorenz, and Christian B. Mendl, "Quantum-Assisted Solution Paths for the Capacitated Vehicle Routing Problem", arXiv:2304.09629, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-05-31 15:27:30) and SAO/NASA ADS (last updated successfully 2023-06-05 13:42:11). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2023-06-05 13:42:08: Encountered the unhandled forward link type postedcontent_cite while looking for citations to DOI 10.22331/q-2021-06-17-479.