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] 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).

[2] 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).

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

[4] Abigail McClain Gomez, Taylor L. Patti, Anima Anandkumar, and Susanne F. Yelin, "Near-term distributed quantum computation using mean-field corrections and auxiliary qubits", Quantum Science and Technology 9 3, 035022 (2024).

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

[6] 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).

[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] 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).

[9] Jason Larkin, Matías Jonsson, Daniel Justice, and Gian Giacomo Guerreschi, "Evaluation of QAOA based on the approximation ratio of individual samples", Quantum Science and Technology 7 4, 045014 (2022).

[10] Stefan H. Sack, Raimel A. Medina, Richard Kueng, and Maksym Serbyn, "Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement", Physical Review A 107 6, 062404 (2023).

[11] 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).

[12] 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).

[13] 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).

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

[15] 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).

[16] 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).

[17] Danylo Lykov, Jonathan Wurtz, Cody Poole, Mark Saffman, Tom Noel, and Yuri Alexeev, "Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm", npj Quantum Information 9, 73 (2023).

[18] Stefan H. Sack and Daniel J. Egger, "Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation", Physical Review Research 6 1, 013223 (2024).

[19] V. Vijendran, Aritra Das, Dax Enshan Koh, Syed M. Assad, and Ping Koy Lam, "An expressive ansatz for low-depth quantum approximate optimisation", Quantum Science and Technology 9 2, 025010 (2024).

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

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

[22] Pranav Chandarana, Pablo Suárez Vieites, Narendra N. Hegade, Enrique Solano, Yue Ban, and Xi Chen, "Meta-learning digitized-counterdiabatic quantum optimization", Quantum Science and Technology 8 4, 045007 (2023).

[23] 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).

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

[25] Daniel J. Egger, Chiara Capecci, Bibek Pokharel, Panagiotis Kl. Barkoutsos, Laurin E. Fischer, Leonardo Guidoni, and Ivano Tavernelli, "Pulse variational quantum eigensolver on cross-resonance-based hardware", Physical Review Research 5 3, 033159 (2023).

[26] Giuseppe Scriva, Nikita Astrakhantsev, Sebastiano Pilati, and Guglielmo Mazzola, "Challenges of variational quantum optimization with measurement shot noise", Physical Review A 109 3, 032408 (2024).

[27] Zichang He, Ruslan Shaydulin, Shouvanik Chakrabarti, Dylan Herman, Changhao Li, Yue Sun, and Marco Pistoia, "Alignment between initial state and mixer improves QAOA performance for constrained optimization", npj Quantum Information 9, 121 (2023).

[28] 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).

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

[30] Benjamin C. B. Symons, David Galvin, Emre Sahin, Vassil Alexandrov, and Stefano Mensa, "A practitioner's guide to quantum algorithms for optimisation problems", Journal of Physics A Mathematical General 56 45, 453001 (2023).

[31] Marvin Bechtold, Johanna Barzen, Frank Leymann, Alexander Mandl, Julian Obst, Felix Truger, and Benjamin Weder, "Investigating the effect of circuit cutting in QAOA for the MaxCut problem on NISQ devices", Quantum Science and Technology 8 4, 045022 (2023).

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

[33] Reuben Tate, Jai Moondra, Bryan Gard, Greg Mohler, and Swati Gupta, "Warm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson's Max-Cut at Low Circuit Depths", Quantum 7, 1121 (2023).

[34] Alexander Mandl, Johanna Barzen, Marvin Bechtold, Frank Leymann, and Karoline Wild, "Amplitude amplification-inspired QAOA: improving the success probability for solving 3SAT", Quantum Science and Technology 9 1, 015028 (2024).

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

[36] Friedrich Wagner, Jonas Nüßlein, and Frauke Liers, "Enhancing Quantum Algorithms for Quadratic Unconstrained Binary Optimization via Integer Programming", arXiv:2302.05493, (2023).

[37] Anita Weidinger, Glen Bigan Mbeng, and Wolfgang Lechner, "Error mitigation for quantum approximate optimization", Physical Review A 108 3, 032408 (2023).

[38] Friedrich Wagner, Daniel J. Egger, and Frauke Liers, "Optimized Noise Suppression for Quantum Circuits", arXiv:2401.06423, (2024).

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

[40] 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).

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

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

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

[44] 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).

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

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

[47] 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", Quantum 7, 1111 (2023).

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

[49] Linus Ekstrom, Hao Wang, and Sebastian Schmitt, "Variational Quantum Multi-Objective Optimization", arXiv:2312.14151, (2023).

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

[51] Julien Gacon, "Scalable Quantum Algorithms for Noisy Quantum Computers", arXiv:2403.00940, (2024).

[52] Taylor L. Patti, Omar Shehab, Khadijeh Najafi, and Susanne F. Yelin, "Markov chain Monte Carlo enhanced variational quantum algorithms", Quantum Science and Technology 8 1, 015019 (2023).

[53] Ken N. Okada, Hirofumi Nishi, Taichi Kosugi, and Yu-ichiro Matsushita, "Systematic study on the dependence of the warm-start quantum approximate optimization algorithm on approximate solutions", Scientific Reports 14, 1167 (2024).

[54] 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).

[55] Libor Caha, Alexander Kliesch, and Robert Koenig, "Twisted hybrid algorithms for combinatorial optimization", Quantum Science and Technology 7 4, 045013 (2022).

[56] M. R. Perelshtein, A. I. Pakhomchik, Ar. A. Melnikov, M. Podobrii, A. Termanova, I. Kreidich, B. Nuriev, S. Iudin, C. W. Mansell, and V. M. Vinokur, "NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization", Quantum 7, 1186 (2023).

[57] Leonardo Ratini, Chiara Capecci, and Leonardo Guidoni, "Optimization strategies in WAHTOR algorithm for quantum computing empirical ansatz: a comparative study", Electronic Structure 5 4, 045006 (2023).

[58] Slimane Thabet, Romain Fouilland, Mehdi Djellabi, Igor Sokolov, Sachin Kasture, Louis-Paul Henry, and Loïc Henriet, "Enhancing Graph Neural Networks with Quantum Computed Encodings", arXiv:2310.20519, (2023).

[59] Samantha V. Barron, Daniel J. Egger, Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz, Matthis Lehmkuehler, and Stefan Woerner, "Provable bounds for noise-free expectation values computed from noisy samples", arXiv:2312.00733, (2023).

[60] 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).

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

[62] Vivek Katial, Kate Smith-Miles, and Charles Hill, "On the Instance Dependence of Optimal Parameters for the Quantum Approximate Optimisation Algorithm: Insights via Instance Space Analysis", arXiv:2401.08142, (2024).

[63] Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon, and Robert Joynt, "Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm", Physical Review Research 5 2, 023147 (2023).

[64] Elias X. Huber, Benjamin Y. L. Tan, Paul R. Griffin, and Dimitris G. Angelakis, "Exponential Qubit Reduction in Optimization for Financial Transaction Settlement", arXiv:2307.07193, (2023).

[65] 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).

[66] Juan Giraldo, José Ossorio, Norha M. Villegas, Gabriel Tamura, and Ulrike Stege, "QPLEX: Realizing the Integration of Quantum Computing into Combinatorial Optimization Software", arXiv:2307.14308, (2023).

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

[68] Mårten Skogh, Oskar Leinonen, Phalgun Lolur, and Martin Rahm, "Accelerating variational quantum eigensolver convergence using parameter transfer", Electronic Structure 5 3, 035002 (2023).

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

[70] Elijah Pelofske, Andreas Bärtschi, and Stephan Eidenbenz, "Short-depth QAOA circuits and quantum annealing on higher-order ising models", npj Quantum Information 10, 30 (2024).

[71] Elijah Pelofske, Georg Hahn, and Hristo Djidjev, "Initial State Encoding via Reverse Quantum Annealing and h-gain Features", arXiv:2303.13748, (2023).

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

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

[74] Yan Jin, Monit Sharma, Hoong Chuin Lau, and Rudy Raymond, "Quantum Relaxation for Solving Multiple Knapsack Problems", arXiv:2404.19474, (2024).

[75] Arne Wulff, Boyang Chen, Matthew Steinberg, Yinglu Tang, Matthias Möller, and Sebastian Feld, "Quantum Computing and Tensor Networks for Laminate Design: A Novel Approach to Stacking Sequence Retrieval", arXiv:2402.06455, (2024).

[76] John Golden, Andreas Bärtschi, Daniel O'Malley, Elijah Pelofske, and Stephan Eidenbenz, "JuliQAOA: Fast, Flexible QAOA Simulation", arXiv:2312.06451, (2023).

[77] Nam H. Le, Milan Sonka, and Fatima Toor, "A Quantum Optimization Method for Geometric Constrained Image Segmentation", arXiv:2310.20154, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-05-21 08:57:57). 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 2024-05-21 08:57:52: Encountered the unhandled forward link type postedcontent_cite while looking for citations to DOI 10.22331/q-2021-06-17-479.