Pith. sign in

REVIEW 4 major objections 4 minor 60 references

Classical solvers and heuristics beat QAOA and quantum annealing on a 250-instance portfolio benchmark, leaving little room for quantum advantage, the paper argues.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 15:49 UTC pith:DKC7DGTC

load-bearing objection Solid negative-result benchmark for quantum portfolio optimization; the central claim holds, but the discretization gap and a couple of benchmark choices need attention. the 4 major comments →

arxiv 2509.17876 v2 pith:DKC7DGTC submitted 2025-09-22 quant-ph math.OC

Quantum Portfolio Optimization: An Extensive Benchmark

Eric Stopfer , Friedrich Wagner This is my paper
classification quant-ph math.OC MSC 81P6891G1090C11
keywords portfolio optimizationquantum annealingQAOAQUBObenchmarkmixed-integer programmingvolatility minimizationquantum advantage
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Building a 250-instance benchmark from real NASDAQ data, this paper argues that the volatility-minimizing portfolio selection problem is not a promising target for near-term quantum optimization. It shows that a commercial mixed-integer programming solver solves every instance, including those with 1,000 assets, to proven optimality in seconds. For the instances small enough to run on quantum hardware (up to about 30 assets), the paper reports that QAOA and quantum annealing produce solutions roughly at the level of random sampling, while a problem-tailored classical heuristic finds better feasible solutions within the same 60-second budget. The authors conclude there is only very limited room for a practical quantum advantage on this variant.

Core claim

The central claim is that for the MinVola variant—minimize portfolio return variance subject to a minimum return and normalized weights—classical methods already settle the problem, so quantum heuristics have no practical edge. A commercial MIP solver solves all benchmark instances to proven optimality in seconds; among heuristics, a greedy problem-specific method dominates. Quantum annealing and QAOA were limited to small instances, and even there their solutions were comparable in quality to random sampling, with the paper attributing the poor performance to the dense all-to-all connectivity of the QUBO, which creates embedding and transpilation overhead. The paper thus sets a high bar for

What carries the argument

The QUBO model of MinVola: the continuous convex program is turned into unconstrained binary optimization by adding quadratic penalties phi*(mu-eps)^2 and psi*(sum omega - 1)^2 (with phi=psi=1000) and discretizing each weight into four binary digits (d=3). This QUBO is what quantum annealing and QAOA actually optimize, and its density is the main mechanism the paper blames for the poor quantum results, since full connectivity forces long qubit chains in annealing and thousands of swap gates in QAOA circuits.

Load-bearing premise

The benchmark uses the continuous MinVola optimum as the reference for all methods, while quantum and QUBO methods solve a coarsely discretized, penalty-weighted version; the paper does not quantify how much of the quantum shortfall is caused by this discretization gap.

What would settle it

Compute, on the same 250 instances, the QUBO solution for finer discretizations (d=4 or 5) and penalty values tuned by search, and compare its best objective to the continuous f_opt; if the gap shrinks to near one, the conclusion of limited quantum advantage would need to be softened. A second check: find a single instance among the 250 that a modern MIP solver cannot prove optimal in seconds, which would break the claim that all instances are easy.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A commercial MIP solver solves this MinVola portfolio variant to proven optimality in seconds even at 1,000 assets, so classical exact methods are not a bottleneck for this problem class.
  • For a fixed 60-second budget, the problem-tailored heuristic beats both QAOA and quantum annealing in solution quality and feasibility on every tested instance size.
  • On the discrete QUBO, quantum annealing and QAOA perform about as well as random sampling; the dense problem structure is identified as the main obstacle.
  • The sampling runtime for quantum annealing does not grow with instance size, implying that only future hardware with better connectivity could change the comparison.
  • Claims of quantum advantage for this variant must be measured against the problem-specific heuristic, not just generic solvers.

Where Pith is reading between the lines

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

  • Because the quantum and QUBO methods optimize a discretized version (d=3, penalty-based) while the reference optimum is the continuous MinVola solution, part of the reported gap could be an artifact of discretization; a finer binary encoding or optimized penalties might close some of the distance to f_opt, at the cost of more qubits.
  • The paper's observation that annealing sample count is independent of problem size suggests an implicit scaling edge for quantum hardware: for very large instances where classical objective evaluations dominate runtime, better-connected future annealers could reverse the ranking even if today's devices lose.
  • The conclusion is tied to the specific MinVola formulation; variants with cardinality constraints, transaction costs, or short-selling limits, which are harder for classical solvers, could still leave room for quantum methods, and the benchmark design could be extended to test that.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper benchmarks quantum annealing and QAOA against classical MIP solvers and heuristics for a minimum-volatility (MinVola) portfolio optimization problem. The test set is generated from real Nasdaq price data and nominally contains 250 instances with up to 1,000 assets. MinVola is encoded as a QUBO via a binary discretization (d=3) and quadratic penalty terms (phi=psi=1000); quantum annealing is run on D-Wave hardware and QAOA on IBM hardware/simulators for instances up to 25 and 20 assets, respectively. Classical methods include Gurobi, SCIP, simulated annealing, steepest descent, tabu search, a problem-specific greedy heuristic, and random sampling. Under a 60-second time limit the paper finds that Gurobi solves all continuous instances to proven optimality in seconds, the problem-specific heuristic outperforms the quantum methods, and QA/QAOA perform roughly at the level of random sampling. The authors conclude that there is very limited room for a quantum advantage for this problem variant.

Significance. If the conclusions hold, the paper is a valuable negative-result benchmark for quantum portfolio optimization. Its strongest and most robust finding is that the continuous MinVola problem can be solved to proven optimality by a commercial MIP solver in seconds, independent of any QUBO encoding. The study also benefits from a real-world instance set, a wide range of classical and quantum methods, a random-sampling control, and public code/data. However, the headline quantitative comparison is weakened by a mismatch between the continuous reference optimum and the discretized penalized objective actually optimized by the QUBO methods; the induced gap is not quantified. Because the central claim is about the limited practical room for quantum advantage, this gap should be measured before the claim can be accepted at face value.

major comments (4)
  1. [Section 6, Eq. (3)-(4), definition of Theta] The primary metric Theta = fm/fopt compares every method against fopt, the proven optimum of the continuous MinVola formulation (1)-(2). All QUBO-based methods, however, minimize the penalized discrete objective (3)-(4) with d=3 and phi=psi=1000. The paper never computes the best objective achievable on this discrete grid, nor checks whether the QUBO optimum is feasible for the original constraints. With only 8 weight levels per asset, the best feasible discrete solution can exceed fopt by a nontrivial amount, and the low feasibility percentages in Fig. 2 indicate that the penalty formulation is not a faithful surrogate for the original constraints. Consequently, part of the observed quantum shortfall relative to fopt may be a discretization/penalty artifact rather than a property of the quantum algorithm. To support the conclusion formulated for the continuous problem, please quantify t
  2. [Section 6, quantum annealing chain strength] The improved chain-strength values cs=3,4,5 for 5-25 assets are selected 'after testing different chain strength parameters for different instance sizes' on the same benchmark instances used for evaluation. This is post hoc tuning on the test set; no separate tuning set or cross-validation is used. Although the default chain-strength results are even worse, so the qualitative conclusion is conservative, the paper's claim to follow good benchmark practice (Ref. [57]) is not fully met. Please either fix chain strengths on a separate tuning set or report results for a range of chain strengths and show the sensitivity.
  3. [Section 6 / Algorithm 1, parameter delta] The problem-specific heuristic is the method that 'consistently outperforms' all other approaches, but its step size delta is never given a numerical value in the main text or in Appendix B. The performance of this greedy heuristic depends critically on delta, and the behavior for ties and infeasible intermediate steps also matters. The code may reveal the value, but the paper itself is not reproducible without reporting delta and the rationale for its choice. Since this heuristic is the strongest classical method in the comparison, the omission directly affects the central claim.
  4. [Section 6, QAOA parameter optimization] The QAOA results are obtained with three parameter heuristics (analytic grid search for p=1, LR-QAOA, and COBYLA on a noiseless simulator), and the authors note that no optimality guarantee for the parameters exists. The observation that more layers worsen performance is attributed to hardware noise and fewer shots. This is a reasonable practical assessment, but the conclusion that QAOA is no better than random sampling is contingent on these particular parameter choices. A sensitivity analysis or a report of the best found parameters would strengthen the claim, and the paper should state this limitation explicitly.
minor comments (4)
  1. [Section 2 / Abstract] The instance count is inconsistent: the test-set generation lists 26 sizes, so 10 instances each gives 260 instances, while the abstract and Section 7 say 250. Please correct.
  2. [Various] Typographical errors: 'variatn', 'sate', 'en par', 'a an state open questions'. Also, the qubit count is given as 4,600 in Section 5 and 4,597 in Section 6; unify.
  3. [Section 2 and 3] The introduction mentions NP-hard optimization, but the three considered variants are convex quadratic programs solvable in polynomial time. The use of 'hardest' in Section 3 is relative to the other two variants; please make this distinction clearer to avoid misleading readers.
  4. [Figure 2 and 3] The text uses 'samples' while Figure 3c uses 'shots'; please unify terminology. Also, specify in the text that the random-sampling baseline draws uniformly from the binary QUBO variables and evaluates the penalized objective; this is important for interpreting the yellow curves.

Circularity Check

0 steps flagged

No circular derivation: the paper's conclusions are empirical measurements against an independently computed MIP optimum; the discrete/continuous gap is a benchmark-validity concern, not circularity.

full rationale

The paper is a computational benchmark. Its central claims—that MinVola is harder for classical solvers and that classical heuristics and MIP outperform QAOA/quantum annealing—are supported by measured runtimes, approximation ratios, and feasibility percentages (Sections 3 and 6). The only derivation in the paper is the QUBO transformation: Eq. (3) adds penalty terms and Eq. (4) discretizes the weights; inserting (4) into (3) yields a QUBO. This is a standard reformulation and is not used to derive the benchmark target. The quality metric Θ=f_m/f_opt uses f_opt computed independently by the MIP solver on the continuous MinVola program (Eqs. 1–2), so the reference value is not constructed from the QUBO or from the quantum methods' outputs. The fact that quantum and QUBO methods optimize a coarse discretized objective (d=3, φ=ψ=1000) while f_opt is the continuous optimum is a legitimate external-validity/fairness concern: the discrete–continuous gap is never quantified, so part of the reported shortfall could be a discretization artifact. But this is not circularity—the paper nowhere equates the discrete QUBO optimum with f_opt, and the quantum shortfall is a measurement, not a consequence of definitions. Likewise, the chain-strength values were tuned on the same instances; this is a benchmark-hygiene issue, not a fitted parameter renamed as a prediction, and the untuned default results are even worse, so the headline conclusion does not depend on the tuned values. There are no load-bearing self-citations: the discretization ansatz is attributed to external works [13,17], QAOA parameter choices to [48,49], and no uniqueness theorem is imported. The paper even notes its own limitations (no optimality guarantee for QAOA parameters; only best-solution comparisons). I therefore find no circular step.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim rests on the QUBO discretization and penalty choices, the runtime comparison protocol, and the instance-generation assumptions. None are derived from first principles, and the quantum annealing chain strengths are fitted to the test set. The heuristic's delta parameter is unspecified, and the discretization error is unquantified.

free parameters (4)
  • Penalty factors phi=psi = 1000
    Chosen by analyzing magnitudes of objective terms (Section 6); not derived from the instance data or a rigorous rule.
  • Discretization depth d = 3
    Chosen as a trade-off between QUBO size and discretization error (Section 6); the discretization error is never quantified.
  • Quantum annealing chain strength cs = 3 for 5-15 assets, 4 for 20 assets, 5 for 25 assets
    Tuned post hoc on the test instances ('After testing different chain strength parameters for different instance sizes, we propose the following improved chain strength values', Section 6). This is fitting to the evaluation set.
  • Heuristic weight increment delta = unspecified
    The problem-specific heuristic's step size delta is a parameter in Algorithm 1 (Appendix B) but its value is not stated in the paper, so the heuristic is not fully specified.
axioms (5)
  • standard math Sample covariance matrix is positive semidefinite
    Invoked to claim convexity of the QCQPs (Section 2: 'The convexity is due to the fact that (sigma_ij) is a sample covariance and thus positive semidefinite by definition (6) in Appendix A').
  • domain assumption Penalty method with finite penalties yields feasible optima
    The QUBO in Eq. (3) replaces hard constraints by penalties; the paper relies on heuristic penalty factors to enforce the return and normalization constraints (Section 4).
  • standard math Adiabatic theorem guarantees ground state for long annealing
    Background for quantum annealing (Section 5); the paper relies on it for correctness of QA as an optimizer, though in practice noise and finite time dominate.
  • domain assumption The 60-second fixed runtime is a fair comparison metric
    All methods are compared by best solution found in 60 s (Section 6); this choice favors methods with fast warm starts and is not justified against other metrics such as time-to-solution for a given quality.
  • domain assumption Generated instances from 2020-2023 Nasdaq data represent realistic portfolio optimization
    Instances are random draws from one market and period (Section 2); this restricts the generality of the 'very limited room for quantum advantage' claim.

pith-pipeline@v1.3.0-alltime-deepseek · 14371 in / 18810 out tokens · 143939 ms · 2026-08-04T15:49:14.962270+00:00 · methodology

0 comments
read the original abstract

Recently, several researchers proposed portfolio optimization as a potential use case for quantum optimization. However, the literature is lacking an extensive benchmark quantifying the potential of quantum computers for portfolio optimization. In this work, we contribute to closing this gap. We provide a computational study, comparing quantum approaches against state-of-the-art classical methods on a meaningful, real-world instance set. In particular, we compare quantum annealing and the quantum approximate optimization algorithm against classical mixed-integer programming, simulated annealing, steepest descent local search, tabu search and a problem-tailored heuristic. We consider a volatility-minimizing variant of portfolio optimization which we show to be more difficult to solve for classical optimizers than return-maximizing or multi-objective formulations. Our benchmark data set comprises 250 instances with up to 1,000 assets from actual stock data. Due to hardware limitation, quantum methods could only be tested for instances with at most 30 assets. The results show that all instances can be solved to proven optimality by mixed-integer programming in the order of seconds. Moreover, the problem-tailored heuristic consistently outperforms quantum approaches in terms of solution quality for fixed runtime. Thus, we conclude that there is only very limited room for a potential quantum advantage for the considered variant of portfolio optimization.

Figures

Figures reproduced from arXiv: 2509.17876 by Eric Stopfer, Friedrich Wagner.

Figure 1
Figure 1. Figure 1: Average solver runtime for different problem variants. We compare SCIP (a) and Gurobi (b) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Summarized computational results. We report the approximation ratio, feasibility percentage and number [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the best performing configurations from our computational study. We report [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 3 canonical work pages

  1. [1]

    Springer, Berlin, Heidelberg, 2018

    Bernhard Korte and Jens Vygen.Combinatorial Optimization: Theory and Algorithms, volume 21 of Algorithms and Combinatorics. Springer, Berlin, Heidelberg, 2018. URL:https://link.springer. com/10.1007/978-3-662-56039-6

    doi:10.1007/978-3-662-56039-6 2018

  2. [2]

    Progress in mathematical programming solvers from 2001 to 2020.EURO Journal on Computational Optimization, 10:100031, 2022.doi:10.1016/j.ejco.2022.100031

    Thorsten Koch, Timo Berthold, Jaap Pedersen, and Charlie Vanaret. Progress in mathematical programming solvers from 2001 to 2020.EURO Journal on Computational Optimization, 10:100031, 2022.doi:10.1016/j.ejco.2022.100031

    arXiv 2001

  3. [3]

    Liebling, Denis Naddef, George L

    Michael Jünger, Thomas M. Liebling, Denis Naddef, George L. Nemhauser, William R. Pulleyblank, Gerhard Reinelt, Giovanni Rinaldi, and Laurence A. Wolsey, editors.50 Years of Integer Program- ming 1958-2008: From the Early Years to the State-of-the-Art. Springer, Berlin, Heidelberg, 2010. URL:http://link.springer.com/10.1007/978-3-540-68279-0

    doi:10.1007/978-3-540-68279-0 1958

  4. [4]

    Christophel, Kati Jarck, Thorsten Koch, Jeff Linderoth, Marco Lübbecke, Hans D

    Ambros Gleixner, Gregor Hendel, Gerald Gamrath, Tobias Achterberg, Michael Bastubbe, Timo Berthold, Philipp M. Christophel, Kati Jarck, Thorsten Koch, Jeff Linderoth, Marco Lübbecke, Hans D. Mittelmann, Derya Ozyurt, Ted K. Ralphs, Domenico Salvagnin, and Yuji Shinano. MI- PLIB 2017: Data-Driven Compilation of the 6th Mixed-Integer Programming Library.Mat...

    doi:10.1007/s12532-020-00194-3 2017

  5. [5]

    Egger, Bruce G

    Amira Abbas, Andris Ambainis, Brandon Augustino, Andreas Bärtschi, Harry Buhrman, Carleton Coffrin, Giorgio Cortiana, Vedran Dunjko, Daniel J. Egger, Bruce G. Elmegreen, Nicola Franco, Filippo Fratini, Bryce Fuller, Julien Gacon, Constantin Gonciulea, Sander Gribling, Swati Gupta, Stuart Hadfield, Raoul Heese, Gerhard Kircher, Thomas Kleinert, Thorsten Ko...

    2024

  6. [6]

    Fekete, Paulina L

    Sabrina Ammann, Maximilian Hess, Debora Ramacciotti, Sándor P. Fekete, Paulina L. A. Goedicke, David Gross, Andreea Lefterovici, Tobias J. Osborne, Michael Perk, Antonio Rotundo, S. E. Skelton, SebastianStiller, andTimodeWolff. RealisticRuntimeAnalysisforQuantumSimplexComputation, November 2023. arXiv:2311.09995 [quant-ph]. URL:http://arxiv.org/abs/2311.09995

    Pith/arXiv arXiv 2023

  7. [7]

    Quantum Comput- ing for Discrete Optimization: A Highlight of Three Technologies, September 2024

    Alexey Bochkarev, Raoul Heese, Sven Jäger, Philine Schiewe, and Anita Schöbel. Quantum Comput- ing for Discrete Optimization: A Highlight of Three Technologies, September 2024. arXiv:2409.01373 [math]. URL:http://arxiv.org/abs/2409.01373

    Pith/arXiv arXiv 2024

  8. [8]

    TameemAlbash andDaniel A. Lidar. Demonstrationof a Scaling Advantage fora Quantum Annealer over Simulated Annealing.Phys. Rev. X, 8(3):031016, July 2018. Publisher: American Physical Society.doi:10.1103/PhysRevX.8.031016

    work page doi:10.1103/physrevx.8.031016 2018

  9. [9]

    McGeoch and Pau Farré

    Catherine C. McGeoch and Pau Farré. Milestones on the Quantum Utility Highway: Quantum Annealing Case Study.ACM Transactions on Quantum Computing, 5(1), December 2023. Place: New York, NY, USA Publisher: Association for Computing Machinery.doi:10.1145/3625307

    work page doi:10.1145/3625307 2023

  10. [10]

    A Quantum Approximate Optimization Al- gorithm, November 2014

    Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A Quantum Approximate Optimization Al- gorithm, November 2014. arXiv:1411.4028 [quant-ph]. URL:http://arxiv.org/abs/1411.4028

    Pith/arXiv arXiv 2014

  11. [11]

    Harrigan, Kevin J

    Matthew P. Harrigan, Kevin J. Sung, Matthew Neeley, Kevin J. Satzinger, Frank Arute, Kunal Arya, Juan Atalaya, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Daniel Eppens, A...

    2021

  12. [12]

    Bernal Neira, Ying Chen, Giorgio Cortiana, Daniel J

    Thorsten Koch, David E. Bernal Neira, Ying Chen, Giorgio Cortiana, Daniel J. Egger, Raoul Heese, Narendra N. Hegade, Alejandro Gomez Cadavid, Rhea Huang, Toshinari Itoko, Thomas Kleinert, Pedro Maciel Xavier, Naeimeh Mohseni, Jhon A. Montanez-Barrera, Koji Nakano, Gi- acomo Nannicini, Corey O’Meara, Justin Pauckert, Manuel Proissl, Anurag Ramesh, Maximili...

    Pith/arXiv arXiv 2025

  13. [13]

    Oberreuter, Riccardo Aiolfi, Luca Asproni, Branislav Roman, and Jürgen Schiefer

    Wolfgang Sakuler, Johannes M. Oberreuter, Riccardo Aiolfi, Luca Asproni, Branislav Roman, and Jürgen Schiefer. A real world test of Portfolio Optimization with Quantum Annealing, March 2023. arXiv:2303.12601 [quant-ph]. URL:http://arxiv.org/abs/2303.12601

    Pith/arXiv arXiv 2023

  14. [14]

    Hybrid Gate-Based and Annealing Quantum Computing for Large-Size Ising Problems, August 2022

    Chen-Yu Liu and Hsi-Sheng Goan. Hybrid Gate-Based and Annealing Quantum Computing for Large-Size Ising Problems, August 2022. arXiv:2208.03283 [quant-ph]. URL:http://arxiv.org/ abs/2208.03283

    Pith/arXiv arXiv 2022

  15. [15]

    Andrist, Grant Salton, Martin J

    Atithi Acharya, Romina Yalovetzky, Pierre Minssen, Shouvanik Chakrabarti, Ruslan Shaydulin, Rudy Raymond, Yue Sun, Dylan Herman, Ruben S. Andrist, Grant Salton, Martin J. A. Schuetz, Helmut G. Katzgraber, and Marco Pistoia. Decomposition Pipeline for Large-Scale Portfolio Optim- ization with Applications to Near-Term Quantum Computing, November 2024. arXi...

    Pith/arXiv arXiv 2024

  16. [16]

    Portfolio Optimisation Using the D-Wave Quantum Annealer

    Frank Phillipson and Harshil Singh Bhatia. Portfolio Optimisation Using the D-Wave Quantum Annealer. In Maciej Paszynski, Dieter Kranzlmüller, Valeria V. Krzhizhanovskaya, Jack J. Dongarra, and Peter M. A. Sloot, editors,Computational Science – ICCS 2021, pages 45–59, Cham, 2021. Springer International Publishing.doi:10.1007/978-3-030-77980-1_4

    work page doi:10.1007/978-3-030-77980-1_4 2021

  17. [17]

    Quantum Portfolio Optimization with Investment Bands and Target Volatility, August 2021

    Samuel Palmer, Serkan Sahin, Rodrigo Hernandez, Samuel Mugel, and Roman Orus. Quantum Portfolio Optimization with Investment Bands and Target Volatility, August 2021. arXiv:2106.06735 [q-fin]. URL:http://arxiv.org/abs/2106.06735

    Pith/arXiv arXiv 2021

  18. [18]

    Reverse Quantum Annealing Approach to Portfolio Op- timization Problems.Quantum Machine Intelligence, 1(1-2):17–30, May 2019

    Davide Venturelli and Alexei Kondratyev. Reverse Quantum Annealing Approach to Portfolio Op- timization Problems.Quantum Machine Intelligence, 1(1-2):17–30, May 2019. arXiv:1810.08584 [quant-ph, q-fin]. URL:http://arxiv.org/abs/1810.08584

    Pith/arXiv arXiv 2019

  19. [19]

    Benchmarking the performance of portfolio optimization with QAOA.Quantum Information Processing, 22(1):25, December 2022

    Sebastian Brandhofer, Daniel Braun, Vanessa Dehn, Gerhard Hellstern, Matthias Hüls, Yanjun Ji, Ilia Polian, Amandeep Singh Bhatia, and Thomas Wellens. Benchmarking the performance of portfolio optimization with QAOA.Quantum Information Processing, 22(1):25, December 2022. arXiv:2207.10555 [quant-ph]. URL:http://arxiv.org/abs/2207.10555

    Pith/arXiv arXiv 2022

  20. [20]

    Portfolio rebalancing experiments using the Quantum Alternating Operator Ansatz, November 2019

    Mark Hodson, Brendan Ruck, Hugh Ong, David Garvin, and Stefan Dulman. Portfolio rebalancing experiments using the Quantum Alternating Operator Ansatz, November 2019. arXiv:1911.05296 [quant-ph]. URL:http://arxiv.org/abs/1911.05296

    Pith/arXiv arXiv 2019

  21. [21]

    Comparative analysis of diverse methodologies for portfolio optimization leveraging quantum annealing techniques, July 2024

    Zhijie Tang, Alex Lu Dou, and Arit Kumar Bishwas. Comparative analysis of diverse methodologies for portfolio optimization leveraging quantum annealing techniques, July 2024. arXiv:2403.02599 [quant-ph]. URL:http://arxiv.org/abs/2403.02599

    Pith/arXiv arXiv 2024

  22. [22]

    Best practices for portfolio optimization by quantum computing, experimented on real quantum devices.Scientific Reports, 13(1):19434, November 2023

    Giuseppe Buonaiuto, Francesco Gargiulo, Giuseppe De Pietro, Massimo Esposito, and Marco Pota. Best practices for portfolio optimization by quantum computing, experimented on real quantum devices.Scientific Reports, 13(1):19434, November 2023. URL:https://www.nature. com/articles/s41598-023-45392-w. 15

    2023

  23. [23]

    Efficient Algorithms for mean-variance portfolio optimization with Hard Real -World Constraints

    Francesco Cesarone, Andrea Scozzari, and Fabio Tardella. Efficient Algorithms for mean-variance portfolio optimization with Hard Real -World Constraints

  24. [24]

    Portfoliooptimizationwithdiscrete simulated annealing, October 2022

    ÁlvaroRubio-García, JuanJoséGarcía-Ripoll, andDiegoPorras. Portfoliooptimizationwithdiscrete simulated annealing, October 2022. arXiv:2210.00807 [cond-mat]. URL:http://arxiv.org/abs/ 2210.00807

    Pith/arXiv arXiv 2022

  25. [25]

    Chang, N

    T.-J. Chang, N. Meade, J.E. Beasley, and Y.M. Sharaiha. Heuristics for cardinality constrained portfolio optimisation.Computers & Operations Research, 27(13):1271–1302, November 2000. URL: https://linkinghub.elsevier.com/retrieve/pii/S030505489900074X

    2000

  26. [26]

    Buhler and Hande Y

    Cassidy K. Buhler and Hande Y. Benson. Efficient Solution of Portfolio Optimization Problems via Dimension Reduction and Sparsification, June 2023. arXiv:2306.12639 [q-fin]. URL:http: //arxiv.org/abs/2306.12639

    Pith/arXiv arXiv 2023

  27. [27]

    Alleviating the quantum Big-M problem.npj Quantum Information, 11(1):125, July 2025

    Edoardo Alessandroni, Sergi Ramos-Calderer, Ingo Roth, Emiliano Traversi, and Leandro Aolita. Alleviating the quantum Big-M problem.npj Quantum Information, 11(1):125, July 2025. URL: https://www.nature.com/articles/s41534-025-01067-0

    2025

  28. [28]

    A Continuous Optimization Approach for the Financial Portfolio Selection under Discrete Asset Choice Constraints, April 2014

    Mahdi Moeini. A Continuous Optimization Approach for the Financial Portfolio Selection under Discrete Asset Choice Constraints, April 2014. arXiv:1404.3286 [cs]. URL:http://arxiv.org/abs/ 1404.3286

    Pith/arXiv arXiv 2014

  29. [29]

    Portfolio selection problems in practice: a comparison between linear and quadratic optimization models.Computational Management Science, 12(3):345–370, July 2015

    Francesco Cesarone, Andrea Scozzari, and Fabio Tardella. Portfolio selection problems in practice: a comparison between linear and quadratic optimization models.Computational Management Science, 12(3):345–370, July 2015. arXiv:1105.3594 [q-fin]. URL:http://arxiv.org/abs/1105.3594

    Pith/arXiv arXiv 2015

  30. [30]

    A Scalable Gradient-Based Optim- ization Framework for Sparse Minimum-Variance Portfolio Selection, May 2025

    Sarat Moka, Matias Quiroz, Vali Asimit, and Samuel Muller. A Scalable Gradient-Based Optim- ization Framework for Sparse Minimum-Variance Portfolio Selection, May 2025. arXiv:2505.10099 [stat]. URL:http://arxiv.org/abs/2505.10099

    Pith/arXiv arXiv 2025

  31. [31]

    URL:https://www.ibm.com/docs/en/ icos/22.1.1?topic=cplex-meet

    IBM ILOG CPLEX Optimization Studio, December 2022. URL:https://www.ibm.com/docs/en/ icos/22.1.1?topic=cplex-meet

    2022

  32. [32]

    ScalingPortfolio Diversification with Quantum Circuit Cutting Techniques, June 2025

    VicenteP.Soloviev, AntonioMárquezRomero, JoshKirsopp, andMichalKrompiec. ScalingPortfolio Diversification with Quantum Circuit Cutting Techniques, June 2025. arXiv:2506.08947 [quant-ph]. URL:http://arxiv.org/abs/2506.08947

    Pith/arXiv arXiv 2025

  33. [33]

    Zi Xuan Loke, Say Leng Goh, Graham Kendall, Salwani Abdullah, and Nasser R. Sabar. Portfolio Optimization Problem: A Taxonomic Review of Solution Methodologies.IEEE Access, 11:33100– 33120, 2023. URL:https://ieeexplore.ieee.org/document/10087257/

    arXiv 2023

  34. [34]

    Portfolio Selection.The Journal of Finance, 7(1):77–91, 1952

    Harry Markowitz. Portfolio Selection.The Journal of Finance, 7(1):77–91, 1952. _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1540-6261.1952.tb01525.x. URL:https:// onlinelibrary.wiley.com/doi/abs/10.1111/j.1540-6261.1952.tb01525.x

    arXiv 1952

  35. [35]

    The interior-point revolution in optimization: History, recent developments, and lasting consequences.Bulletin of the American Mathematical Society, 42(1):39–56, September 2004

    Margaret Wright. The interior-point revolution in optimization: History, recent developments, and lasting consequences.Bulletin of the American Mathematical Society, 42(1):39–56, September 2004. URL:https://www.ams.org/bull/2005-42-01/S0273-0979-04-01040-7/

    2004

  36. [36]

    URL:https://finance.yahoo.com/quote/ %5EIXIC/

    NASDAQ Composite (^IXIC) Charts, Data & News. URL:https://finance.yahoo.com/quote/ %5EIXIC/

  37. [37]

    URL:https://finance

    Yahoo Finance - Stock Market Live, Quotes, Business & Finance News. URL:https://finance. yahoo.com/

  38. [38]

    The SCIP Optimization Suite 9.0

    Suresh Bolusani, Mathieu Besancon, Ksenia Bestuzheva, Antonia Chmiela, Joao Dionısio, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Mohammed Ghannam, Ambros Gleixner, Chris- toph Graczyk, Katrin Halbig, Ivo Hedtke, Alexander Hoen, and Christopher Hojny. The SCIP Optimization Suite 9.0

  39. [39]

    URL:https://www.gurobi

    The Leader in Decision Intelligence Technology - Gurobi Optimization. URL:https://www.gurobi. com/. 16

  40. [40]

    Katzgraber, Wolfgang Lechner, Hidetoshi Nishimori, and William D

    Philipp Hauke, Helmut G. Katzgraber, Wolfgang Lechner, Hidetoshi Nishimori, and William D. Oliver. Perspectives of quantum annealing: Methods and implementations.Reports on Progress in Physics, 83(5):054401, May 2020. arXiv:1903.06559 [quant-ph]. URL:http://arxiv.org/abs/ 1903.06559

    Pith/arXiv arXiv 2020

  41. [41]

    Adiabatic quantum computation.Reviews of Modern Physics, 90(1), 2018.doi: 10.1103/RevModPhys.90.015002

    Tameem Albash. Adiabatic quantum computation.Reviews of Modern Physics, 90(1), 2018.doi: 10.1103/RevModPhys.90.015002

    doi:10.1103/revmodphys.90.015002 2018

  42. [42]

    Elsevier, 1961

    Albert Messiah.Quantum Mechanics. Elsevier, 1961. Google-Books-ID: VR93vUk8d_8C

    1961

  43. [43]

    URL:https://www.dwavequantum.com/ solutions-and-products/systems/

    The Advantage™Quantum Computer | D-Wave. URL:https://www.dwavequantum.com/ solutions-and-products/systems/

  44. [44]

    Minor-Embedding in Adiabatic Quantum Computation: I

    Vicky Choi. Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem, April 2008. arXiv:0804.4884 [quant-ph]. URL:http://arxiv.org/abs/0804.4884

    Pith/arXiv arXiv 2008

  45. [45]

    Minor-embedding in adiabatic quantum computation: II

    Vicky Choi. Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Information Processing, 10(3):343–353, June 2011. arXiv:1001.3116 [quant-ph]. URL: http://arxiv.org/abs/1001.3116

    Pith/arXiv arXiv 2011

  46. [46]

    URL:https://dwave-systemdocs

    Samplers — dwave-system 0.9.12 documentation. URL:https://dwave-systemdocs. readthedocs.io/en/latest/reference/samplers.html#dwavecliquesampler

  47. [47]

    Macready, and Aidan Roy

    Jun Cai, William G. Macready, and Aidan Roy. A practical heuristic for finding graph minors, June

  48. [48]

    Expectation values from the single-layer quantum approximate optimization algorithm on Ising problems.Quantum Science and Technology, 7(4):045036, October 2022

    Asier Ozaeta, Wim Van Dam, and Peter L McMahon. Expectation values from the single-layer quantum approximate optimization algorithm on Ising problems.Quantum Science and Technology, 7(4):045036, October 2022. URL:https://iopscience.iop.org/article/10.1088/2058-9565/ ac9013

    doi:10.1088/2058-9565/ 2022

  49. [49]

    J. A. Montanez-Barrera and Kristel Michielsen. Towards a universal QAOA protocol: Evidence of quantum advantage in solving combinatorial optimization problems, May 2024. arXiv:2405.09169 [quant-ph]. URL:http://arxiv.org/abs/2405.09169

    Pith/arXiv arXiv 2024

  50. [50]

    URL:https://docs.scipy.org/doc/ scipy/reference/optimize.minimize-cobyla.html

    minimize(method=’COBYLA’) — SciPy v1.15.3 Manual. URL:https://docs.scipy.org/doc/ scipy/reference/optimize.minimize-cobyla.html

  51. [51]

    URL:https://qiskit.github.io/qiskit-aer/stubs/qiskit_ aer.AerSimulator.html

    AerSimulator - Qiskit Aer 0.16.1. URL:https://qiskit.github.io/qiskit-aer/stubs/qiskit_ aer.AerSimulator.html

  52. [52]

    URL:https://quantum.ibm.com/

    IBM Quantum. URL:https://quantum.ibm.com/

  53. [53]

    original-date: 2019-08-02T10:10:41Z

    dwavesystems/dwave-greedy, September 2024. original-date: 2019-08-02T10:10:41Z. URL:https: //github.com/dwavesystems/dwave-greedy

    2024

  54. [54]

    Page Version ID: 1311714758

    Simulated annealing, September 2025. Page Version ID: 1311714758. URL:https://en.wikipedia. org/w/index.php?title=Simulated_annealing&oldid=1311714758

    2025

  55. [55]

    URL:https://docs.dwavequantum.com/en/latest/ ocean/api_ref_samplers/index.html

    dwave-samplers — Python documentation. URL:https://docs.dwavequantum.com/en/latest/ ocean/api_ref_samplers/index.html

  56. [56]

    original-date: 2018-08-28T02:14:30Z

    dwavesystems/dwave-tabu, September 2024. original-date: 2018-08-28T02:14:30Z. URL:https: //github.com/dwavesystems/dwave-tabu

    2024

  57. [57]

    Benchmarking Quantum Computers: Towards a Standard Performance Evaluation Approach, July 2024

    Arturo Acuaviva, David Aguirre, Rubén Peña, and Mikel Sanz. Benchmarking Quantum Computers: Towards a Standard Performance Evaluation Approach, July 2024. arXiv:2407.10941 [quant-ph]. URL:http://arxiv.org/abs/2407.10941

    arXiv 2024

  58. [58]

    URL:https://github.com/stopfereric/portfolio_opt_ benchmark

    stopfereric/portfolio_opt_benchmark. URL:https://github.com/stopfereric/portfolio_opt_ benchmark

  59. [59]

    URL:https://github.com/ ranaroussi/yfinance

    yfinance: Download market data from Yahoo! Finance API. URL:https://github.com/ ranaroussi/yfinance. 17

  60. [2014]

    URL:http://arxiv.org/abs/1406.2741

    arXiv:1406.2741 [quant-ph]. URL:http://arxiv.org/abs/1406.2741

    Pith/arXiv arXiv