Improving Variational Quantum Optimization using CVaR

Panagiotis Kl. Barkoutsos; Giacomo Nannicini; Anton Robert; Ivano Tavernelli; Stefan Woerner

doi:10.22331/q-2020-04-20-256

Improving Variational Quantum Optimization using CVaR

Panagiotis Kl. Barkoutsos¹, Giacomo Nannicini², Anton Robert^1,3, Ivano Tavernelli¹, and Stefan Woerner¹

¹IBM Research – Zurich
²IBM T.J. Watson Research Center
³École Normale Supérieure, PSL University, Paris

Published:	2020-04-20, volume 4, page 256
Eprint:	arXiv:1907.04769v3
Doi:	https://doi.org/10.22331/q-2020-04-20-256
Citation:	Quantum 4, 256 (2020).

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Abstract

Hybrid quantum/classical variational algorithms can be implemented on noisy intermediate-scale quantum computers and can be used to find solutions for combinatorial optimization problems. Approaches discussed in the literature minimize the expectation of the problem Hamiltonian for a parameterized trial quantum state. The expectation is estimated as the sample mean of a set of measurement outcomes, while the parameters of the trial state are optimized classically. This procedure is fully justified for quantum mechanical observables such as molecular energies. In the case of classical optimization problems, which yield diagonal Hamiltonians, we argue that aggregating the samples in a different way than the expected value is more natural. In this paper we propose the Conditional Value-at-Risk as an aggregation function. We empirically show -- using classical simulation as well as quantum hardware -- that this leads to faster convergence to better solutions for all combinatorial optimization problems tested in our study. We also provide analytical results to explain the observed difference in performance between different variational algorithms.

► BibTeX data

@article{Barkoutsos2020improving,
  doi = {10.22331/q-2020-04-20-256},
  url = {https://doi.org/10.22331/q-2020-04-20-256},
  title = {Improving {V}ariational {Q}uantum {O}ptimization using {CV}a{R}},
  author = {Barkoutsos, Panagiotis Kl. and Nannicini, Giacomo and Robert, Anton and Tavernelli, Ivano and Woerner, Stefan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {256},
  month = apr,
  year = {2020}
}

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[102] Zhaolong Huang, Qiting Li, Junling Zhao, and Meimei Song, "Variational Quantum Algorithm Applied to Collision Avoidance of Unmanned Aerial Vehicles", Entropy 24 11, 1685 (2022).

[103] Thomas Lubinski, Carleton Coffrin, Catherine McGeoch, Pratik Sathe, Joshua Apanavicius, David Bernal Neira, and Quantum Economic Development Consortium(QED-C) Collaboration, "Optimization Applications as Quantum Performance Benchmarks", ACM Transactions on Quantum Computing 5 3, 1 (2024).

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[109] Andrew Jordan Siciliano and Zheng Wang, "Generating three-dimensional genome structures with a variational quantum algorithm", Briefings in Bioinformatics 26 6, bbaf663 (2025).

[110] Yagnik Chatterjee, Eric Bourreau, and Marko J. Rančić, "Solving various NP-hard problems using exponentially fewer qubits on a quantum computer", Physical Review A 109 5, 052441 (2024).

[111] Yousef Hindy, Jessica Pointing, Meltem Tolunay, Sreeram Venkatarao, Mario Motta, and Joseph A. Latone, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 681 (2023) ISBN:979-8-3503-4323-6.

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

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[115] Anbang Wang, Dunbo Cai, Yu Zhang, Yangqing Huang, Xiangyang Feng, and Zhihong Zhang, "Surrogate modeling via factorization machine and Ising model with enhanced higher-order interaction learning", Physical Review Research 8 1, 013037 (2026).

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[117] Zijiang Yan, Hao Zhou, Jianhua Pei, Aryan Kaushik, Hina Tabassum, and Ping Wang, ICC 2025 - IEEE International Conference on Communications 6088 (2025) ISBN:979-8-3315-0521-9.

[118] Huy Phuc Nguyen Ha, Viet Hung Nguyen, and Anh Son Ta, Lecture Notes in Computer Science 14753 , 268 (2024) ISBN:978-3-031-62911-2.

[119] Ruslan Shaydulin and Alexey Galda, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 291 (2021) ISBN:978-1-6654-1691-7.

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[121] Qiting Li, Zhaolong Huang, Weiwei Jiang, Zikang Tang, and Meimei Song, "Quantum Algorithms Using Infeasible Solution Constraints for Collision-Avoidance Route Planning", IEEE Transactions on Consumer Electronics 71 1, 2186 (2025).

[122] Qingqing YU and Rong JIN, "Improved Quantum Approximate Optimization Algorithm Based on Conditional Value-at-Risk for Portfolio Optimization", IEICE Transactions on Information and Systems E108.D 7, 727 (2025).

[123] Yu Zhang, Xueyun Cheng, Fei Ding, Pengcheng Zhu, Zhijin Guan, and Hui Gu, "A quantum circuit parameterized decomposition method based on local synthesis", Quantum Information Processing 24 10, 333 (2025).

[124] Renata Wong and Weng-Long Chang, "Fast quantum algorithm for protein structure prediction in hydrophobic-hydrophilic model", Journal of Parallel and Distributed Computing 164, 178 (2022).

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[126] Felix Gemeinhardt, Antonio Garmendia, Manuel Wimmer, Benjamin Weder, and Frank Leymann, "Quantum Combinatorial Optimization in the NISQ Era: A Systematic Mapping Study", ACM Computing Surveys 56 3, 1 (2024).

[127] Mona Sobhani, Yahui Chai, Tobias Hartung, and Karl Jansen, "Variational quantum eigensolver approach to prime factorization on IBM's noisy intermediate scale quantum computer", Physical Review A 111 4, 042413 (2025).

[128] Robert J. Banks, Dan E. Browne, and P.A. Warburton, "Rapid quantum approaches for combinatorial optimisation inspired by optimal state-transfer", Quantum 8, 1253 (2024).

[129] Ajinkya Borle, Vincent Elfving, and Samuel J. Lomonaco, "Quantum approximate optimization for hard problems in linear algebra", SciPost Physics Core 4 4, 031 (2021).

[130] Haomu Yuan, Christopher K Long, Hugo V Lepage, and Crispin H W Barnes, "Quantifying the advantages of applying quantum approximate algorithms to portfolio optimisation", Quantum Science and Technology 11 2, 025034 (2026).

[131] Simone Cantori, David Vitali, and Sebastiano Pilati, "Supervised learning of random quantum circuits via scalable neural networks", Quantum Science and Technology 8 2, 025022 (2023).

[132] Zoé Verchère, Sourour Elloumi, and Andrea Simonetto, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 19 (2023) ISBN:979-8-3503-4323-6.

[133] Giancarlo Martínez Salirrosas, Jinglun Gao, Arthur Yu, and Anish Ravi Verma, "Market informed portfolio optimization methods with hybrid quantum computing", Review of Financial Economics 43 1, 62 (2025).

[134] Anton Robert, Panagiotis Kl. Barkoutsos, Stefan Woerner, and Ivano Tavernelli, "Resource-efficient quantum algorithm for protein folding", npj Quantum Information 7 1, 38 (2021).

[135] Shengbin Wang, Peng Wang, Guihui Li, Shubin Zhao, Dongyi Zhao, Jing Wang, Yuan Fang, Menghan Dou, Yongjian Gu, Yu-Chun Wu, and Guo-Ping Guo, "Variational quantum eigensolver with linear depth problem-inspired ansatz for solving portfolio optimization in finance", Science China Information Sciences 68 8, 180504 (2025).

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[146] Tavis Bennett, Lyle Noakes, and Jingbo Wang, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 31 (2024) ISBN:979-8-3315-4137-8.

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[154] Xuan-Bac Nguyen, Hoang-Quan Nguyen, Hugh Churchill, Samee U. Khan, and Khoa Luu, "Quantum visual feature encoding revisited", Quantum Machine Intelligence 6 2, 61 (2024).

[155] Daniel J. Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Simonetto, Stefan Woerner, and Elena Yndurain, "Quantum Computing for Finance: State-of-the-Art and Future Prospects", IEEE Transactions on Quantum Engineering 1, 1 (2020).

[156] Gines Carrascal, Guillermo Botella, Alberto del Barrio, and David Kremer, "A Bayesian-network-based quantum procedure for failure risk analysis", EPJ Quantum Technology 10 1, 13 (2023).

[157] Qilin Zheng, Miaomiao Yu, Pingyu Zhu, Yan Wang, Weihong Luo, and Ping Xu, "Solving the subset sum problem by the quantum Ising model with variational quantum optimization based on conditional values at risk", Science China Physics, Mechanics & Astronomy 67 8, 280311 (2024).

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[159] Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Differentiable quantum architecture search", Quantum Science and Technology 7 4, 045023 (2022).

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[161] Yahui Chai and Alice Di Tucci, "Optimizing QUBO on a quantum computer by mimicking imaginary time evolution", New Journal of Physics 28 6, 064508 (2026).

[162] Simone Cantori and Sebastiano Pilati, "Challenges and opportunities in the supervised learning of quantum circuit expectation values", Physical Review E 112 1, 015304 (2025).

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[165] Abhinav Anand, Jonathan Romero, Matthias Degroote, and Alán Aspuru‐Guzik, "Noise Robustness and Experimental Demonstration of a Quantum Generative Adversarial Network for Continuous Distributions", Advanced Quantum Technologies 4 5, 2000069 (2021).

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[167] Yahui Chai, Karl Jansen, Stefan Kühn, Tim Schwägerl, and Tobias Stollenwerk, "Warm start of variational quantum algorithms for quadratic unconstrained binary optimization problems", EPJ Quantum Technology 13 1, 9 (2026).

[168] Vicente P. Soloviev, Concha Bielza, and Pedro Larrañaga, "Quantum approximate optimization algorithm for Bayesian network structure learning", Quantum Information Processing 22 1, 19 (2022).

[169] Claudio Gambella and Andrea Simonetto, "Multiblock ADMM Heuristics for Mixed-Binary Optimization on Classical and Quantum Computers", IEEE Transactions on Quantum Engineering 1, 1 (2020).

[170] Ramin Ayanzadeh, Narges Alavisamani, Poulami Das, and Moinuddin Qureshi, Proceedings of the 28th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2 311 (2023) ISBN:9781450399166.

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[174] Ryan LaRose, Eleanor Rieffel, and Davide Venturelli, "Mixer-phaser Ansätze for quantum optimization with hard constraints", Quantum Machine Intelligence 4 2, 17 (2022).

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[176] Dimitris Alevras, Mihir Metkar, Takahiro Yamamoto, Vaibhaw Kumar, Triet Friedhoff, Jae-Eun Park, Mitsuharu Takeori, Mariana LaDue, Wade Davis, and Alexey Galda, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 488 (2024) ISBN:979-8-3315-4137-8.

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[178] Sabina Drăgoi, Alberto Baiardi, and Daniel J. Egger, "Approximate quadratization of high-order Hamiltonians for combinatorial quantum optimization", Physical Review Research 8 2, 023159 (2026).

[179] Akshay Uttarkar and Vidya Niranjan, "A comparative insight into peptide folding with quantum CVaR-VQE algorithm, MD simulations and structural alphabet analysis", Quantum Information Processing 23 2, 48 (2024).

[180] T. Miki, D. Tsukayama, R. Okita, M Shimada, and J. Shirakashi, 2022 IEEE 22nd International Conference on Nanotechnology (NANO) 303 (2022) ISBN:978-1-6654-5225-0.

[181] Samuel Fernández-Lorenzo, Diego Porras, and Juan José García-Ripoll, "Hybrid quantum–classical optimization with cardinality constraints and applications to finance", Quantum Science and Technology 6 3, 034010 (2021).

[182] Srinivasarao Thota, C.B. Krishna, Navaneet Ojha, Parcha Kalyani, and Thulasi Bikku, "On A Quantum-Fractional Framework for Modeling Tobacco Smoking Dynamics", Procedia Computer Science 282, 449 (2026).

[183] Matteo Puviani, Sangkha Borah, Remmy Zen, Jan Olle, and Florian Marquardt, "Non-Markovian Feedback for Optimized Quantum Error Correction", Physical Review Letters 134 2, 020601 (2025).

[184] Hiroshi Yano, Yudai Suzuki, Kohei Itoh, Rudy Raymond, and Naoki Yamamoto, "Efficient Discrete Feature Encoding for Variational Quantum Classifier", IEEE Transactions on Quantum Engineering 2, 1 (2021).

[185] R. Thilagavathy, M. Gayathri, G. K. Sandhia, and M. Pushpalatha, Advances in Computer and Electrical Engineering 367 (2024) ISBN:9798369311684.

[186] 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", Nature Computational Science 4 11, 865 (2024).

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[188] Lee Braine, Daniel J. Egger, Jennifer Glick, and Stefan Woerner, "Quantum Algorithms for Mixed Binary Optimization Applied to Transaction Settlement", IEEE Transactions on Quantum Engineering 2, 1 (2021).

[189] Linghua Zhu, Ho Lun Tang, George S. Barron, F. A. Calderon-Vargas, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer", Physical Review Research 4 3, 033029 (2022).

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