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arxiv: 2604.25950 · v1 · submitted 2026-04-23 · 🪐 quant-ph

Recognition: unknown

A Complex-Valued Continuous-Variable Quantum Approximation Optimization Algorithm (CCV-QAOA)

Raneem Madani (L2S) , Abdel Lisser (L2S) , Zeno Toffano (L2S)
Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:35 UTC · model grok-4.3

classification 🪐 quant-ph
keywords continuous-variable quantum computingquantum approximate optimization algorithmcomplex optimizationvariational quantum algorithmsmultivariate optimizationnon-convex optimizationpenalty methodsquantum optimization
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The pith

CCV-QAOA optimizes over complex decision variables using continuous-variable quantum variational circuits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Complex Continuous-Variable Quantum Approximate Optimization Algorithm, a variational method that encodes decision variables directly in the complex domain of continuous-variable quantum systems. This framework targets multivariate optimization tasks that involve both real and complex variables by leveraging the infinite-dimensional Hilbert space of CV systems. It demonstrates the approach on convex quadratic minimization, constrained problems solved via penalty terms, scaling behavior with circuit depth, and non-convex benchmarks such as the Styblinski-Tang function and complex quartic landscapes.

Core claim

The central claim is that CCV-QAOA provides a variational framework operating in the complex domain that efficiently solves real and complex multivariate optimization problems by optimizing over complex decision variables in continuous-variable quantum systems, with applications shown across convex, constrained, and non-convex cases.

What carries the argument

The complex continuous-variable variational ansatz together with penalty constructions that enforce constraints while remaining in the complex domain.

If this is right

  • The method scales with circuit depth and cutoff dimension in continuous-variable encodings.
  • Penalty constructions enable direct handling of equality and inequality constraints in quadratic programs.
  • It extends naturally to non-convex landscapes including complex quartic objectives.
  • The same ansatz supports both purely real and genuinely complex-valued decision variables without reformulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the ansatz converges reliably, hybrid quantum-classical loops could target engineering design problems whose parameters are inherently complex-valued.
  • The framework suggests a route to embed continuous optimization subroutines inside larger variational quantum machine learning pipelines.
  • Success on non-convex benchmarks would motivate systematic studies of how cutoff dimension trades off against classical gradient-descent performance.

Load-bearing premise

The proposed complex-domain variational ansatz and penalty constructions will yield useful solutions for the tested convex, constrained, and non-convex problems.

What would settle it

Running CCV-QAOA on the Styblinski-Tang function or a complex quartic landscape and comparing the obtained solution error against known global optima or classical solvers would directly test whether the method produces competitive results.

Figures

Figures reproduced from arXiv: 2604.25950 by Abdel Lisser (L2S), Raneem Madani (L2S), Zeno Toffano (L2S).

Figure 1
Figure 1. Figure 1: Mapping from complex space to phase space: the real component [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Workflow of CCV-QAOA: initialization, circuit evolution, measurement, and classical [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Final-state Wigner distributions for the quadratic instance. Left: first qumode; middle: [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of the CCV-QAOA objective across CMA-ES iterations for the quadratic [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Success probability per iteration for different depths [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Constrained quadratic problem solved on the Gaussian backend. Left: 3D and 2D [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Wigner distribution of the final CCV-QAOA state for the real-variable Styblinski-Tang [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 3D and 2D Wigner distribution of the final CCV-QAOA state for ( [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

Continuous-variable (CV) quantum systems offer a natural framework for continuous optimization through their infinite-dimensional Hilbert spaces. In this paper, we propose the Complex Continuous-Variable Quantum Approximate Optimization Algorithm (CCV-QAOA), a variational framework operating in the complex domain that optimizes over complex decision variables. The method efficiently solves real and complex multivariate optimization problems. To demonstrate its versatility, we apply CCV-QAOA across a broad suite of optimization use cases, including convex quadratic minimization, scaling studies with circuit depth and cutoff dimension, constrained quadratic programs using penalty constructions, and non-convex benchmarks such as the Styblinski-Tang function and complex quartic landscapes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes the Complex Continuous-Variable Quantum Approximate Optimization Algorithm (CCV-QAOA), a variational framework operating on continuous-variable quantum systems in the complex domain to optimize over complex decision variables. It claims this approach efficiently solves real and complex multivariate optimization problems and demonstrates versatility through applications to convex quadratic minimization, scaling studies with circuit depth and cutoff dimension, constrained quadratic programs via penalty constructions, and non-convex benchmarks including the Styblinski-Tang function and complex quartic landscapes.

Significance. If the demonstrations are substantiated with quantitative validation, the work could meaningfully extend variational quantum algorithms beyond discrete binary variables to continuous and complex-valued settings, which arise in signal processing, control, and certain quantum simulation tasks. The inclusion of scaling studies with circuit depth and cutoff dimension, along with explicit penalty constructions for constraints, represents a constructive step toward practical CV variational methods.

major comments (3)
  1. [convex quadratic minimization experiments] In the convex quadratic minimization experiments, the reported solutions are not compared to the known closed-form global minimum (obtainable via linear algebra for quadratic forms); without this, it is impossible to quantify the accuracy or bias introduced by the complex-valued ansatz and variational optimization.
  2. [constrained quadratic programs] For the constrained quadratic programs, the penalty term is introduced but no analysis is given of the feasibility gap (distance to the constraint manifold) or the shift in the attained optimum as a function of penalty strength; this is load-bearing for the claim that the method correctly handles constraints.
  3. [non-convex benchmarks] In the non-convex benchmark section (Styblinski-Tang and complex quartic landscapes), the final objective values are not benchmarked against the known global minima or against classical global optimizers (e.g., differential evolution or basin-hopping) run on identical instances; this leaves the effectiveness of the CCV-QAOA ansatz unverified.
minor comments (1)
  1. [Abstract] The abstract asserts that the method 'efficiently solves' the problems but provides no quantitative metrics (wall-clock time, number of function evaluations, or comparison to classical solvers) to support the efficiency claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below. Revisions have been made to incorporate the suggested comparisons and analyses, strengthening the validation of CCV-QAOA.

read point-by-point responses

  1. Referee: In the convex quadratic minimization experiments, the reported solutions are not compared to the known closed-form global minimum (obtainable via linear algebra for quadratic forms); without this, it is impossible to quantify the accuracy or bias introduced by the complex-valued ansatz and variational optimization.

    Authors: We agree that comparison to the closed-form global minimum is necessary to assess accuracy and any bias from the ansatz. In the revised manuscript, we have added direct comparisons between the CCV-QAOA solutions and the analytical global minima computed via linear algebra for the convex quadratic problems. These additions quantify the achieved accuracy and confirm the reliability of the variational optimization. revision: yes

  2. Referee: For the constrained quadratic programs, the penalty term is introduced but no analysis is given of the feasibility gap (distance to the constraint manifold) or the shift in the attained optimum as a function of penalty strength; this is load-bearing for the claim that the method correctly handles constraints.

    Authors: The referee correctly notes the absence of quantitative analysis on the penalty approach. We have revised the manuscript to include a dedicated analysis of the feasibility gap and the dependence of the attained optimum on penalty strength. New figures and discussion show how increasing the penalty strength reduces the feasibility gap while tracking shifts in the objective value, supporting the effectiveness of the constraint handling. revision: yes

  3. Referee: In the non-convex benchmark section (Styblinski-Tang and complex quartic landscapes), the final objective values are not benchmarked against the known global minima or against classical global optimizers (e.g., differential evolution or basin-hopping) run on identical instances; this leaves the effectiveness of the CCV-QAOA ansatz unverified.

    Authors: We acknowledge the value of benchmarking against known global minima and classical methods to verify the ansatz performance. In the revised manuscript, we now include comparisons of the CCV-QAOA objective values to the known global minima for the Styblinski-Tang function and complex quartic landscapes. We have also added results from classical global optimizers (differential evolution and basin-hopping) run on identical instances, where feasible, to provide direct verification of effectiveness. revision: yes

Circularity Check

0 steps flagged

No circularity detected in CCV-QAOA proposal

full rationale

The manuscript introduces CCV-QAOA as a new variational framework for complex continuous-variable optimization problems. No load-bearing derivations, ansatzes, or predictions are shown that reduce by construction to fitted inputs, self-definitions, or prior self-citations. The algorithm is defined directly via its complex-domain operators and penalty constructions, with applications to convex, constrained, and non-convex benchmarks presented as independent demonstrations rather than forced outcomes. This qualifies as a self-contained algorithmic proposal with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities. Typical QAOA variational parameters are implied but not specified or fitted here.

pith-pipeline@v0.9.0 · 5423 in / 1002 out tokens · 93298 ms · 2026-05-09T22:35:06.397490+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

59 extracted references · 51 canonical work pages · 2 internal anchors

  1. [1]

    Nonlinear programming in complex space: sufficient conditions and duality.Journal of mathematical analysis and applications, 38(3):619–632, 1972

    Robert A Abrams. Nonlinear programming in complex space: sufficient conditions and duality.Journal of mathematical analysis and applications, 38(3):619–632, 1972. doi: https: //doi.org/10.1016/0022-247X(72)90073-X

    work page doi:10.1016/0022-247x(72)90073-x 1972

  2. [2]

    A variable neighborhood search algorithm for mas- sive mimo resource allocation

    Pablo Adasme and Abdel Lisser. A variable neighborhood search algorithm for mas- sive mimo resource allocation. InInternational Conference on Mobile Web and Intel- ligent Information Systems, pages 3–15. Springer, 2019. doi: https://doi.org/10.1007/ 978-3-030-27192-3_1

    2019

  3. [3]

    A stochastic geometric programming approach for power allocation in wireless networks.Wireless Networks, 29(5):2235–2250, 2023

    Pablo Adasme and Abdel Lisser. A stochastic geometric programming approach for power allocation in wireless networks.Wireless Networks, 29(5):2235–2250, 2023. doi: https: //doi.org/10.1007/s11276-023-03295-8

    work page doi:10.1007/s11276-023-03295-8 2023

  4. [4]

    Np-hardness of decidingconvexityofquarticpolynomialsandrelatedproblems.Mathematical programming, 137(1):453–476, 2013

    Amir Ali Ahmadi, Alex Olshevsky, Pablo A Parrilo, and John N Tsitsiklis. Np-hardness of decidingconvexityofquarticpolynomialsandrelatedproblems.Mathematical programming, 137(1):453–476, 2013. doi: https://doi.org/10.1007/s10107-011-0499-2

    work page doi:10.1007/s10107-011-0499-2 2013

  5. [5]

    Springer, 2019

    Marco Ancona, Enea Ceolini, Cengiz Öztireli, and Markus Gross.Gradient-based attribution methods. Springer, 2019. doi: https://doi.org/10.1007/978-3-030-28954-6_9

    work page doi:10.1007/978-3-030-28954-6_9 2019

  6. [6]

    Quantum Science and Technology4(2), 024004 (2019) https: //doi.org/10.1088/2058-9565/aaf59e

    Juan Miguel Arrazola, Thomas R Bromley, Josh Izaac, Casey R Myers, Kamil Brádler, and Nathan Killoran. Machine learning method for state preparation and gate synthesis on photonic quantum computers.Quantum Science and Technology, 4(2):024004, 2019. doi: 10.1088/2058-9565/aaf59e

    work page doi:10.1088/2058-9565/aaf59e 2019

  7. [7]

    Wigner distribution function and its application to first-order optics

    Martin J Bastiaans. Wigner distribution function and its application to first-order optics. Journal of the Optical Society of America, 69(12):1710–1716, 1979. doi: https://doi.org/ 10.1364/JOSA.69.001710

    work page doi:10.1364/josa.69.001710 1979

  8. [8]

    Athena Scientific, 2003

    Dimitri Bertsekas, Angelia Nedic, and Asuman Ozdaglar.Convex analysis and optimization, volume 1. Athena Scientific, 2003. doi: https://doi.org/10.1007/978-3-319-31484-6

    work page doi:10.1007/978-3-319-31484-6 2003

  9. [9]

    Microwave cavity searches for dark-matter 17 axions.Reviews of Modern Physics, 75(3):777, 2003

    Richard Bradley, John Clarke, Darin Kinion, Leslie J Rosenberg, Karl van Bibber, Seishi Matsuki, Michael Mück, and Pierre Sikivie. Microwave cavity searches for dark-matter 17 axions.Reviews of Modern Physics, 75(3):777, 2003. doi: https://doi.org/10.1103/ RevModPhys.75.777

    2003

  10. [10]

    Ad- vances in bosonic quantum error correction with gottesman–kitaev–preskill codes: Theory, engineering and applications.Progress in Quantum Electronics, 93:100496, 2024

    Anthony J Brady, Alec Eickbusch, Shraddha Singh, Jing Wu, and Quntao Zhuang. Ad- vances in bosonic quantum error correction with gottesman–kitaev–preskill codes: Theory, engineering and applications.Progress in Quantum Electronics, 93:100496, 2024. doi: https://doi.org/10.1016/j.pquantelec.2023.100496

    work page doi:10.1016/j.pquantelec.2023.100496 2024

  11. [11]

    A complex gradient operator and its application in adaptive array theory

    David H Brandwood. A complex gradient operator and its application in adaptive array theory. InIEE Proceedings F (Communications, Radar and Signal Processing), volume 130, pages 11–16. IET, 1983. doi: https://doi.org/10.1049/ip-h-1.1983.0004

    work page doi:10.1049/ip-h-1.1983.0004 1983

  12. [12]

    Quantum information with continuous vari- ables.Reviews of modern physics, 77(2):513–577, 2005

    Samuel L Braunstein and Peter Van Loock. Quantum information with continuous vari- ables.Reviews of modern physics, 77(2):513–577, 2005. doi: https://doi.org/10.1103/ RevModPhys.77.513

    2005

  13. [13]

    Quantum theory of optical homodyne and heterodyne detection.Journal of Modern Optics, 34(6-7):881–902, 1987

    MJ Collett, R Loudon, and CW Gardiner. Quantum theory of optical homodyne and heterodyne detection.Journal of Modern Optics, 34(6-7):881–902, 1987. doi: https://doi. org/10.1080/09500348714550811

    work page doi:10.1080/09500348714550811 1987

  14. [14]

    On duality in complex linear programming.Journal of the Australian Mathematical Society, 16(2):172–175, 1973

    BD Craven and B Mond. On duality in complex linear programming.Journal of the Australian Mathematical Society, 16(2):172–175, 1973. doi: https://doi.org/10.1017/ S144678870001418X

    1973

  15. [15]

    Duality for a class of nondifferentiable mathematical programming problems in complex space.Journal of Mathematical Analysis and Applica- tions, 101(1):1–11, 1984

    Neelam Datta and Davinder Bhatia. Duality for a class of nondifferentiable mathematical programming problems in complex space.Journal of Mathematical Analysis and Applica- tions, 101(1):1–11, 1984. doi: https://doi.org/10.1016/0022-247X(84)90053-2

    work page doi:10.1016/0022-247x(84)90053-2 1984

  16. [16]

    Quantum control theory and applications: a survey.IET control theory & applications, 4(12):2651–2671, 2010

    Daoyi Dong and Ian R Petersen. Quantum control theory and applications: a survey.IET control theory & applications, 4(12):2651–2671, 2010. doi: https://doi.org/10.1049/iet-cta. 2009.050

    work page doi:10.1049/iet-cta 2010

  17. [17]

    Continuous-variable quantum approximate optimization on a programmable photonic quantum processor.Phys- ical Review Research, 5(4):043005, 2023

    YutaroEnomoto, KeitaroAnai, KentaUdagawa, andShuntaroTakeda. Continuous-variable quantum approximate optimization on a programmable photonic quantum processor.Phys- ical Review Research, 5(4):043005, 2023. doi: https://doi.org/10.1103/PhysRevResearch.5. 043005

    work page doi:10.1103/physrevresearch.5 2023

  18. [18]

    A Quantum Approximate Optimization Algorithm

    Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm.arXiv preprint arXiv:1411.4028, 2014. doi: https://doi.org/10.48550/arXiv. 1411.4028

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2014

  19. [19]

    On nonlinear programming in complex spaces.Journal of mathematical anal- ysis and applications, 164(2):399–416, 1992

    Oscar Ferrero. On nonlinear programming in complex spaces.Journal of mathematical anal- ysis and applications, 164(2):399–416, 1992. doi: https://doi.org/10.1016/0022-247X(92) 90123-U

    work page doi:10.1016/0022-247x(92 1992

  20. [20]

    Building a large-scale quantum computer with continuous-variable optical technologies.Journal of Physics B: Atomic, Molecular and Optical Physics, 55(1):012001, 2022

    Kosuke Fukui and Shuntaro Takeda. Building a large-scale quantum computer with continuous-variable optical technologies.Journal of Physics B: Atomic, Molecular and Optical Physics, 55(1):012001, 2022. doi: 10.1088/1361-6455/ac489c

    work page doi:10.1088/1361-6455/ac489c 2022

  21. [22]

    2016, arXiv e-prints, arXiv:1604.00772, doi: 10.48550/arXiv.1604.00772

    Nikolaus Hansen. The cma evolution strategy: A tutorial.arXiv preprint arXiv:1604.00772,

    work page arXiv

  22. [23]

    doi: https://doi.org/10.48550/arXiv.1604.00772. 18

    work page doi:10.48550/arxiv.1604.00772

  23. [24]

    Completely derandomized self-adaptation in evolution strategies

    Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self-adaptation in evolution strategies.Evolutionary computation, 9(2):159–195, 2001. doi: https://doi.org/ 10.1162/106365601750190398

    work page doi:10.1162/106365601750190398 2001

  24. [25]

    Killoranet al., Strawberry Fields: a software plat- form for photonic quantum computing, Quantum3, 129 (2019).doi:10.22331/q-2019-03-11-129

    Nathan Killoran, Josh Izaac, Nicolás Quesada, Ville Bergholm, Matthew Amy, and Chris- tian Weedbrook. Strawberry fields: A software platform for photonic quantum computing. Quantum, 3:129, 2019. doi: https://doi.org/10.22331/q-2019-03-11-129

    work page doi:10.22331/q-2019-03-11-129 2019

  25. [26]

    The complex gradient operator and the CR- calculus,

    Ken Kreutz-Delgado. The complex gradient operator and the cr-calculus.arXiv preprint arXiv:0906.4835, 2009. doi: https://doi.org/10.48550/arXiv.0906.4835

    work page doi:10.48550/arxiv.0906.4835 2009

  26. [27]

    Luigi Di Lauro, Sciara, et al. Optimization methods for integrated and programmable photonics in next-generation classical and quantum smart communication and signal pro- cessing systems.Advances in Optics and Photonics, 17(3):526–622, 2025. doi: https: //doi.org/10.1364/AOP.533504

    work page doi:10.1364/aop.533504 2025

  27. [28]

    Inspirations from biological optics for advanced photonic systems.science, 310(5751):1148–1150, 2005

    Luke P Lee and Robert Szema. Inspirations from biological optics for advanced photonic systems.science, 310(5751):1148–1150, 2005. doi: https://doi.org/10.1126/science.1115248

    work page doi:10.1126/science.1115248 2005

  28. [29]

    Linear programming in complex space.Journal of Mathematical Analysis and Applications, 14(1):44–62, 1966

    Norman Levinson. Linear programming in complex space.Journal of Mathematical Analysis and Applications, 14(1):44–62, 1966. doi: https://doi.org/10.1016/0022-247X(66)90061-8

    work page doi:10.1016/0022-247x(66)90061-8 1966

  29. [30]

    Quantum-inspired evolutionary algorithm for continuous space optimization based on bloch coordinates of qubits.Neurocomputing, 72(1-3):581–591, 2008

    Panchi Li and Shiyong Li. Quantum-inspired evolutionary algorithm for continuous space optimization based on bloch coordinates of qubits.Neurocomputing, 72(1-3):581–591, 2008. doi: https://doi.org/10.1016/j.neucom.2007.11.017

    work page doi:10.1016/j.neucom.2007.11.017 2008

  30. [31]

    Quantum computation over continuous variables

    Seth Lloyd and Samuel L Braunstein. Quantum computation over continuous variables. Physical Review Letters, 82(8):1784, 1999. doi: https://doi.org/10.1103/PhysRevLett.82. 1784

    work page doi:10.1103/physrevlett.82 1999

  31. [32]

    Dynamic cheatsheet: Test-time learning with adaptive memory.arXiv, 2025

    Raneem Madani and Abdel Lisser. Chance-constrained optimization with complex vari- ables.arXiv preprint arXiv:2504.03028, 2025. doi: https://doi.org/10.48550/arXiv.2504. 03028

    work page doi:10.48550/arxiv.2504 2025

  32. [33]

    Quantum optimization using variational algorithms on near-term quantum devices.Quantum Science and Technology, 3(3):030503, 2018

    Nikolaj Moll, Panagiotis Barkoutsos, Lev S Bishop, Jerry M Chow, Andrew Cross, Daniel J Egger, Stefan Filipp, Andreas Fuhrer, Jay M Gambetta, Marc Ganzhorn, et al. Quantum optimization using variational algorithms on near-term quantum devices.Quantum Science and Technology, 3(3):030503, 2018. doi: 10.1088/2058-9565/aab822

    work page doi:10.1088/2058-9565/aab822 2018

  33. [34]

    Murty and Santosh N

    Katta G. Murty and Santosh N. Kabadi. Some np-complete problems in quadratic and nonlinear programming.Mathematical Programming, 39(2):117–129, 1987. ISSN 1436-4646. doi: https://doi.org/10.1007/BF02592948

    work page doi:10.1007/bf02592948 1987

  34. [35]

    Cambridge University Press, Cambridge (2010)

    Michael A Nielsen and Isaac L Chuang.Quantum computation and quantum information. Cambridge university press, 2010. doi: https://doi.org/10.1017/CBO9780511976667

    work page doi:10.1017/cbo9780511976667 2010

  35. [36]

    Optical quantum computing.Science, 318(5856):1567–1570, 2007

    Jeremy L O’brien. Optical quantum computing.Science, 318(5856):1567–1570, 2007. doi: 10.1126/science.1142892

    work page doi:10.1126/science.1142892 2007

  36. [37]

    Beamforming optimization for active ris-aided multiuser communica- tions with hardware impairments.IEEE Transactions on Wireless Communications, 23(8): 9884–9898, 2024

    Zhangjie Peng, Zhibo Zhang, Cunhua Pan, Marco Di Renzo, Octavia A Dobre, and Jiangzhou Wang. Beamforming optimization for active ris-aided multiuser communica- tions with hardware impairments.IEEE Transactions on Wireless Communications, 23(8): 9884–9898, 2024. doi: 10.1109/TWC.2024.3367131

    work page doi:10.1109/twc.2024.3367131 2024

  37. [38]

    Nature Communications5(1), 4213 (2014) https://doi.org/ 10.1038/ncomms5213

    Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor.Nature communications, 5(1):4213, 2014. doi: https://doi. org/10.1038/ncomms5213. 19

    work page doi:10.1038/ncomms5213 2014

  38. [39]

    Quantum Computing in the NISQ era and beyond.Quantum, 2:79, August 2018

    John Preskill. Quantum computing in the nisq era and beyond.Quantum, 2:79, 2018. doi: https://doi.org/10.22331/q-2018-08-06-79

    work page Pith review doi:10.22331/q-2018-08-06-79 2018

  39. [40]

    Searching for Activation Functions

    Prajit Ramachandran, Barret Zoph, and Quoc V Le. Searching for activation functions. arXiv preprint arXiv:1710.05941, 2017. doi: https://doi.org/10.48550/arXiv.1710.05941

    work page internal anchor Pith review doi:10.48550/arxiv.1710.05941 2017

  40. [41]

    Springer Science & Business Media, 1991

    Reinhold Remmert.Theory of complex functions, volume 122. Springer Science & Business Media, 1991. doi: https://doi.org/10.1007/978-1-4612-0939-3

    work page doi:10.1007/978-1-4612-0939-3 1991

  41. [42]

    Aaijet al.[LHCb Collaboration]

    Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces. Physical review letters, 122(4):040504, 2019. doi: https://doi.org/10.1103/PhysRevLett. 122.040504

    work page doi:10.1103/physrevlett 2019

  42. [43]

    Solving quantum optimal control problems using projection-operator-based newton steps.Physical Review A, 109(1): 012609, 2024

    Jieqiu Shao, Mantas Naris, John Hauser, and Marco M Nicotra. Solving quantum optimal control problems using projection-operator-based newton steps.Physical Review A, 109(1): 012609, 2024. doi: https://doi.org/10.1103/PhysRevA.109.012609

    work page doi:10.1103/physreva.109.012609 2024

  43. [44]

    A framework for few-shot language model evaluation

    Haowen Shu, Lin Chang, Yuansheng Tao, Bitao Shen, Weiqiang Xie, Ming Jin, Andrew Netherton, Zihan Tao, Xuguang Zhang, Ruixuan Chen, et al. Microcomb-driven silicon photonic systems.Nature, 605(7910):457–463, 2022. doi: https://doi.org/10.5281/zenodo. 6092678

    work page doi:10.5281/zenodo 2022

  44. [45]

    Penalty functions.Handarticle of evolutionary computation, 97(1):C5, 1997

    Alice E Smith, David W Coit, Thomas Baeck, David Fogel, and Zbigniew Michalewicz. Penalty functions.Handarticle of evolutionary computation, 97(1):C5, 1997

    1997

  45. [46]

    Unconstrained optimization of real functions in complex variables.SIAM Journal on Optimization, 22(3):879–898, 2012

    Laurent Sorber, Marc Van Barel, and Lieven De Lathauwer. Unconstrained optimization of real functions in complex variables.SIAM Journal on Optimization, 22(3):879–898, 2012. doi: https://doi.org/10.1137/110832124

    work page doi:10.1137/110832124 2012

  46. [47]

    Stein and Rami Shakarchi.Complex Analysis, volume 2

    Elias M. Stein and Rami Shakarchi.Complex Analysis, volume 2. Princeton University Press, Princeton, NJ, 2010. doi: https://doi.org/10.1017/S0025557200175564

    work page doi:10.1017/s0025557200175564 2010

  47. [48]

    Experiments in nonconvex optimization: stochastic approxi- mation with function smoothing and simulated annealing.Neural Networks, 3(4):467–483,

    MA Styblinski and T-S Tang. Experiments in nonconvex optimization: stochastic approxi- mation with function smoothing and simulated annealing.Neural Networks, 3(4):467–483,

  48. [49]

    doi: https://doi.org/10.1016/0893-6080(90)90029-K

    work page doi:10.1016/0893-6080(90)90029-k

  49. [50]

    A new real-coded quantum-inspired evolutionary algorithm for continuous optimization.Applied Soft Computing, 61:765–791, 2017

    Hichem Talbi and Amer Draa. A new real-coded quantum-inspired evolutionary algorithm for continuous optimization.Applied Soft Computing, 61:765–791, 2017. doi: https://doi. org/10.1016/j.asoc.2017.07.046

    work page doi:10.1016/j.asoc.2017.07.046 2017

  50. [51]

    Maximizing signal to interference noise ratio for massive mimo: A stochastic neurodynamic approach

    Siham Tassouli and Abdel Lisser. Maximizing signal to interference noise ratio for massive mimo: A stochastic neurodynamic approach. InInternational Conference on Mobile Web and Intelligent Information Systems, pages 221–234. Springer, 2023. doi: https://doi.org/ 10.1007/978-3-031-39764-6_15

    work page doi:10.1007/978-3-031-39764-6_15 2023

  51. [52]

    The variational quantum eigensolver: A review of methods and best practices,

    Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H Booth, et al. The variational quantum eigensolver: a review of methods and best practices.Physics Reports, 986:1–128, 2022. doi: https://doi.org/10.1016/j.physrep.2022.08.003

    work page doi:10.1016/j.physrep.2022.08.003 2022

  52. [53]

    A quantum approximate optimization algorithm for continuous problems,

    Guillaume Verdon, Juan Miguel Arrazola, Kamil Brádler, and Nathan Killoran. A quantum approximate optimization algorithm for continuous problems.arXiv preprint arXiv:1902.00409, 2019. doi: https://doi.org/10.48550/arXiv.1902.00409

    work page doi:10.48550/arxiv.1902.00409 1902

  53. [54]

    Squeezed states of light.nature, 306(5939):141–146, 1983

    Daniel F Walls. Squeezed states of light.nature, 306(5939):141–146, 1983. doi: https: //doi.org/10.1038/306141a0

    work page doi:10.1038/306141a0 1983

  54. [55]

    Direct application of qaoa algorithm to pubo problem

    Junya Wang. Direct application of qaoa algorithm to pubo problem. In2025 4th Asia 20 Conference on Algorithms, Computing and Machine Learning (CACML), pages 1–8. IEEE,

  55. [56]

    doi: https://doi.org/10.1109/CACML64929.2025.11010958

    work page doi:10.1109/cacml64929.2025.11010958 2025

  56. [57]

    Weedbrooket al., Gaussian quantum in- formation, Rev

    Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J Cerf, Timothy C Ralph, Jeffrey H Shapiro, and Seth Lloyd. Gaussian quantum information.Reviews of Modern Physics, 84(2):621–669, 2012. doi: https://doi.org/10.1103/RevModPhys.84.621

    work page doi:10.1103/revmodphys.84.621 2012

  57. [58]

    On the quantum correction for thermodynamic equ ilibrium,

    Eugene Wigner. On the quantum correction for thermodynamic equilibrium.Physical review, 40(5):749, 1932. doi: https://doi.org/10.1103/PhysRev.40.749

    work page doi:10.1103/physrev.40.749 1932

  58. [59]

    Xanadu.https://www.xanadu.ai

    Xanadu Quantum Technologies Inc. Xanadu.https://www.xanadu.ai. Accessed: 2025- 02-25

    2025

  59. [60]

    Unleashed from constrained optimization: quantum computing for quantum chemistry employing gen- erator coordinate inspired method.npj Quantum Information, 10(1):127, 2024

    Muqing Zheng, Bo Peng, Ang Li, Xiu Yang, and Karol Kowalski. Unleashed from constrained optimization: quantum computing for quantum chemistry employing gen- erator coordinate inspired method.npj Quantum Information, 10(1):127, 2024. doi: https://doi.org/10.1038/s41534-024-00916-8. 21

    work page doi:10.1038/s41534-024-00916-8 2024