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REVIEW 4 major objections 7 minor 25 references

Co-optimization of codon usage and mRNA secondary structure using quantum computing

T0 review · 4 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A hybrid quantum-classical algorithm co-optimizes mRNA codon choice and folding, and runs end-to-end on a 127-qubit Eagle processor.

desk verdict Real-hardware hybrid mRNA co-optimization loop is a genuine first, but the quantum solver's QUBO parameters are left unspecified, making the 'optimal' claim unverifiable as written. read the letter →

arxiv 2507.18817 v1 pith:L3TLSGFS submitted 2025-07-24 quant-ph

Dimitris Alevras , Mihir Metkar , Triet Friedhoff , Jae-Eun Park , Mariana LaDue , Vaibhaw Kumar , Wade Davis , Alexey Galda This is my paper · ORCID
classification quant-ph
keywords mRNAdesigncodonoptimizationsecondarystructurepredictionhybridquantum-classicalalgorithmCVaRvariationalquadraticbinarysuperconductingquantumprocessoradaptationindex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An mRNA sequence has to be easy for the ribosome to translate and able to fold into a stable structure, and the two goals fight because both come from the same nucleotide string. This paper tries to settle that fight with one optimization loop: a classical solver picks a codon for each amino acid, a variational quantum subroutine predicts how the resulting RNA folds, and a composite score combining codon adaptation index (CAI) and minimum free energy (MFE) tells both pieces how to improve. The central claim is that this loop is not only simulable but executable on today's hardware: the full workflow, including the quantum fold-prediction step, ran on a 127-qubit Eagle processor and produced the same optimal mRNA constructs for the tested short sequences as the simulator did. That matters because co-optimized mRNAs are the kind of constructs used in therapeutics, where stability in the vial and translation in the cell both depend on getting the sequence right.

What carries the argument

The mechanism that carries the argument is the quadratic binary program of Eq. (7), in which every decision variable is a 'quartet'—two consecutive stacked base pairs—and the objective sums nearest-neighbor free energies from the Turner database, rewards adjacent stacks, and penalizes folds that end in a UA pair. This program is solved by a CVaR variational quantum algorithm that concentrates sampling on low-energy bitstrings, while the outer optimization is a Nelder-Mead loop over the weights θc, θp, θr of the codon-selection linear program, which CPLEX solves exactly and which encodes GC content, rarity, and repeat penalties. The two halves are coupled by the composite objective f = α·CAI + MFE with α = −0.5, so each candidate codon choice is judged by how well the sequence it produces folds.

What would settle it

Enumerate all synonymous codon assignments for a short amino-acid sequence, compute the Eq. (7) score and the ViennaRNA MFE for each, and find one sequence where the two disagree about which fold is best; if the outer optimizer then selects a codon choice by the Eq. (7) ranking, the surrogate is steering the design away from the thermodynamically optimal structure.

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Extended reading notes

Core claim

The core discovery is that codon usage and secondary structure can be co-optimized in a closed variational loop that survives contact with real quantum hardware. For the six-amino-acid sequence TLPKAD, the loop converged to the 18-nucleotide sequence ACUCUGCCGAAGGCAGAC with CAI 0.718 and MFE −8.100, matching the classically predicted dot-bracket fold, and the hardware run reproduced the simulated optimization trajectory; a 13-amino-acid sequence sampled on a second Eagle processor returned the optimal fold with the highest frequency. Across ten independent runs on sequences of 18 to 48 nucleotides, the method found the classically consistent structure 70–100% of the time. The paper reads this as evidence that the hybrid workflow, with its CVaR variational quantum solver for secondary structure, is a feasible design loop on current superconducting processors.

Load-bearing premise

The whole scheme depends on the simplified fold-scoring formula of Eq. (7), with its two unstated reward and penalty constants, ranking candidate mRNA structures in the same order as true minimum free energy for every sequence the optimizer generates.

Editorial extensions

If this is right

  • The end-to-end loop runs on current 127-qubit hardware, so the quantum secondary-structure subroutine is demonstrated outside pure simulation for short mRNAs.
  • For the tested sequences, the optimizer's chosen folds agree with classical predictions in 70–100% of independent runs, so the hybrid loop is consistent, not just convergent.
  • Caching previously evaluated sequences cuts redundant quantum calls, letting the Nelder-Mead loop finish in 12–27 iterations on the tested cases.
  • The tunable parameter α lets a designer shift the balance between codon optimality and structural stability within the same framework.
  • Scaling to therapeutically sized mRNAs will face circuit-depth and optimizer-efficiency limits, pointing toward modular decomposition and surrogate models as planned next steps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Eq. (7) ever ranks folds differently from ViennaRNA's MFE on a sequence the optimizer visits, the outer loop would be optimizing toward a distorted target; a direct check would compare the two rankings over all synonymous codon assignments for one amino-acid sequence.
  • Because the reward r and penalty p in Eq. (7) are never given numerical values, the reported hardware agreement is not independently reproducible without those constants; publishing them would let others rerun the exact experiment.
  • The same loop could be pointed at other RNA-design objectives—avoiding G-quadruplex motifs, cell-type-specific codon tables, or repeat penalties—by swapping the codon metric while keeping the quantum fold subroutine.
  • The biological payoff is asserted but not measured: synthesizing the hardware-selected mRNAs and measuring expression and half-life would test whether the co-optimized sequences actually behave better than codon-only designs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The manuscript introduces a hybrid quantum-classical workflow for co-optimizing mRNA codon usage and secondary structure. A classical Nelder-Mead loop updates a parameter vector θ = (θc, θp, θr) that weights GC content, rarity, and repeat penalties in a codon-selection ILP (Eq. 6). The ILP solution yields a nucleotide sequence; the sequence's CAI is computed directly, and its secondary structure is predicted by a CVaR-based variational quantum solver solving the QUBO in Eq. (7). The resulting dot-bracket structure is scored with ViennaRNA's eval_structure, and the combined objective is f = α·CAI + MFE (Eq. 1). The authors report simulation results on seven amino-acid sequences up to 16 residues, with success rates of 70–100%, and hardware runs on IBM's 127-qubit Eagle processors for the TLPKAD example and for a 13-residue sequence sampled on ibm_fez. They conclude that end-to-end execution of the hybrid algorithm on real quantum hardware is feasible and that the hardware results reproduce simulation.

Significance. If fully substantiated, this is a useful systems-integration milestone: it demonstrates that a variational quantum subroutine can be embedded in a closed-loop sequence-design workflow and executed on a 127-qubit processor, with results matching simulation for short sequences. The decomposition is sensible: an exact classical solver handles the easy codon-selection ILP, while the quantum solver targets the more complex secondary-structure QUBO, and the external ViennaRNA evaluator provides a classical check on the predicted fold. The paper also includes a multi-sequence simulation table and a hardware demonstration, which is more than a purely theoretical proposal. However, the current evidence is not yet sufficient to establish the central feasibility claim in a reproducible form: key QUBO parameters are missing, the success metric is undefined, and the simulator-versus-hardware comparison lacks quantitative detail. The results are therefore of interest to the quantum-computing-for-biology community, but the manuscript needs substantial revision before the claims can be verified.

major comments (4)
  1. [Section II, Eq. (7)] The objective function in Eq. (7) contains two undetermined scalar parameters: the stacking reward r and the terminal-pair penalty p. No numerical values, ranges, units, or scaling conventions are provided anywhere in the text, tables, figures, or references, and the index sets QS(qi) and QUA are not specified precisely enough to reconstruct the second and third sums. Consequently Eq. (7) does not define a concrete optimization problem, and the claim that hardware sampling reproduced simulation cannot be independently verified. This is load-bearing because the reported MFE values and final sequences depend on the structure returned by the solver that minimizes Eq. (7). Please specify r and p explicitly (for example, values from the Turner/NNDB parameters or from Ref. [19]) and clarify the semantics of the QS(qi) and QUA sets.
  2. [Section III-A, Table I] The 'success rate' reported in Table I is not defined, and no test is reported comparing the Eq. (7) optimum with the true ViennaRNA MFE structure for the sequences generated during optimization. Since Function 1 scores the quantum-solver structure with eval_structure, the outer loop's sequence choices are meaningful only if the Eq. (7) optimum coincides with the MFE optimum (or if the objective-value discrepancy is negligible). The text states this coincidence only for the final TLPKAD fold. Please define what 'success' counts across the ten runs, report per-sequence agreement between the surrogate optimum and the ViennaRNA MFE optimum, and quantify any impact of disagreement on the composite objective f.
  3. [Section III-B] The hardware claim rests on one TLPKAD trajectory and one sampling run on ibm_fez, but the paper provides no quantitative comparison between simulator and hardware output distributions, no fidelity metric, and no error bars. Figure 2 does not include a simulator baseline, axis labels, or a description of what the 'objective value' bins represent. Without these, the statement that 'the optimal structure was sampled with the highest frequency' on hardware, and the broader assertion that hardware reproduced simulation, are not supported by the displayed data. Please add side-by-side simulator and hardware distributions, a quantitative agreement measure (e.g., variational distance or Hellinger distance), and details on shot counts and any post-selection.
  4. [Section III-A and Table I] The reported final results for the TLPKAD example are internally inconsistent: the text and Fig. 1 give the final nucleotide sequence as ACUCUGCCGAAGGCAGAC with 27 iterations and 106 function evaluations, while Table I lists ACUCUGCCUAAGGCGGAC with 24 iterations and 99 function evaluations. Since this example is the cornerstone of the hardware demonstration, these discrepancies must be reconciled and the correct numbers reported consistently in the text, figure, and table.
minor comments (7)
  1. [Section II] The symbol n is used for both the number of amino acids and the nucleotide sequence length in the secondary-structure model; using a distinct symbol (e.g., L) for the nucleotide length would avoid ambiguity.
  2. [Section III-B] The phrase 'Nshots = 213 bitstrings' presumably means 2^13 = 8192 bitstrings; please clarify the shot count and how the raw samples were converted to objective values.
  3. [Section II, Eq. (7)] The p-term writes qi(1−qj) with qj ∈ QUA, which is confusing because QUA is described as a set of stacked quartets; please state directly which variables incur the terminal (U,A) penalty rather than using an implicit complement.
  4. [Figure 2] Please add axis labels, units, and a legend; the claim that the optimal structure is the most frequent cannot be checked from the current figure.
  5. [Table I] Add a caption or text defining 'success rate' and reporting the number of runs; the 'ten independent runs' is mentioned only in the text and not in the table caption.
  6. [Section II, Eq. (6)] Eq. (6) describes θ as the 'initial vector of variational parameters', but θ is the quantity optimized by Nelder-Mead; use 'optimization variable' or 'variational parameter' consistently.
  7. [Section II] The relationship between Eq. (6)'s GC/rarity/repeat objective and the CAI term in Eq. (1) is not discussed; a sentence explaining why minimizing the rarity penalty with weights θp is aligned with maximizing CAI would help the reader understand the design.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the co-optimization claim is anchored to external ViennaRNA and classical-structure checks; the self-cited QUBO subroutine is validated rather than assumed, and the unspecified r,p is a reproducibility gap, not a circular reduction.

full rationale

The derivation chain is not circular. The composite objective f(θ) = α CAI(θ) + MFE(θ) is evaluated using external references: CAI uses human codon frequencies from python_codon_tables, and MFE is computed with ViennaRNA's eval_structure, an independent thermodynamic engine. The quantum secondary-structure subroutine is imported from the authors' prior QCE 2024 paper (Ref. [19]), which is a self-citation, but the current paper does not treat that subroutine as an unverified premise: for the TLPKAD example it states that 'the predicted folding matches the classically computed optimal structure,' and for hardware execution it compares the sampled objective distribution against simulator results. Thus the central claims (end-to-end hardware execution and simulator/hardware agreement) are checked against external or independently reproducible references rather than reduced to the subroutine's own definition. The reward r and penalty p in Eq. (7) are never numerically specified, and the 'success rate' in Table I is not defined; these are reproducibility and correctness gaps, not circularity, because no claimed output is shown to equal an input by construction. The self-citation of Ref. [19] is methodologically load-bearing, but it is not the sole support for the paper's validation claims, so it does not make the derivation circular.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or conserved quantities. Its contributions are algorithmic: two hand-chosen weights (alpha and the QUBO penalties r, p) and per-instance optimization weights (theta) that are tuned rather than derived. These are counted as free parameters because the final codon choices and conclusions depend on them.

free parameters (3)
  • alpha (balancing weight) = -0.5
    Weight between CAI and MFE in Eq. (1), chosen by hand; the paper gives no derivation for this value.
  • theta_c, theta_p, theta_r (codon ILP weights) = (0.955, 6.476, 7.234) for TLPKAD
    Weights for GC, rarity, and repeat penalties in Eq. (6), tuned per instance by Nelder-Mead; these values change the final codon choice.
  • r (stacking reward) and p (UA-pair penalty) = not specified in paper
    Parameters in Eq. (7) that shape the energy landscape of the structure QUBO; without their values the structure objective is not reproducible from the text.
assumptions (6)
  • domain assumption Nearest-neighbor energy parameters from the Turner/NNDB model are valid for scoring mRNA base-pair stacks.
    Invoked in Section II via Ref. [22] to assign energy contributions e_qi to quartet variables in Eq. (7).
  • domain assumption Human codon usage frequencies from the python_codon_tables library reflect translation efficiency.
    Used in Eq. (4) for the rarity penalty, following Refs. [14] and [16].
  • standard math The quartet-based binary formulation from Ref. [20] correctly encodes feasible non-crossing RNA secondary structures.
    Stated at the start of Section II: 'Following the formulation in Ref. [20]'; the constraints in Eq. (8) are accepted without proof here.
  • domain assumption ViennaRNA's eval_structure computes the true minimum free energy for a given dot-bracket structure.
    Function 1 in Section II uses MFE(s, dotb) via eval_structure as the final objective currency.
  • standard math CPLEX solves the codon ILP in Eq. (6) exactly.
    The paper states codon optimization 'is not NP-hard and admits efficient classical solutions' and relies on CPLEX for exact solutions.
  • domain assumption A single variational loop with Nelder-Mead converges to a useful local optimum of the combined objective for these instances.
    The results report run-to-run success rates below 100% for several sequences, indicating the loop is not guaranteed to converge to the global optimum.

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Cite this review

Pith. "Pith review of Co-optimization of codon usage and mRNA secondary structure using quantum computing." pith.science (2026). https://pith.science/paper/L3TLSGFS

@misc{pith2026250718817,
  author       = {Pith},
  title        = {Pith review of: Co-optimization of codon usage and mRNA secondary structure using quantum computing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3TLSGFS}},
  note         = {Machine review of arXiv:2507.18817}
}
read the original abstract

Co-optimizing mRNA sequences for both codon optimality and secondary structure is crucial for producing stable and efficacious mRNA therapeutics. Codon optimization, which adjusts nucleotide sequences to enhance translational efficiency, inherently influences mRNA secondary structure - a key determinant of molecular stability both in-vial and in-cell. Because both properties are governed by the same underlying sequence, optimizing one directly impacts the other. To address this interdependence, we introduce a novel variational framework that simultaneously optimizes codon usage and secondary structure. Our method employs a dual-objective function that balances the codon adaptation index (CAI) and minimum free energy (MFE), incorporating variational parameters for codon selection. Leveraging a hybrid quantum-classical computational strategy and building on prior advancements in quantum algorithms for secondary structure prediction, we effectively navigate this complex optimization space. We demonstrate the feasibility of executing this end-to-end workflow on real quantum hardware, using IBM's 127-qubit Eagle processor. We validate our approach through both simulations and quantum hardware experiments on sequences of up to 30 nucleotides. These results highlight the potential of our framework to accelerate the design of optimal mRNA constructs for therapeutic and research applications.

Figures

Figures reproduced from arXiv: 2507.18817 by the authors.

Figure 1
Figure 1. Workflow of the hybrid mRNA optimization algorithm. Starting from an amino acid sequence, a classical optimizer iteratively updates codon selection [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Sample distribution of objective values from [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Works this paper leans on

25 extracted references · 23 canonical work pages

  1. [19]

    mrna secondary structure prediction using utility-scale quantum computers,

    D. Alevras, M. Metkar, T. Yamamoto, V . Kumar, T. Friedhoff, J.- E. Park, M. Takeori, M. LaDue, W. Davis, and A. Galda, “mrna secondary structure prediction using utility-scale quantum computers,” in 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) , vol. 1. IEEE, 2024, pp. 488–499

    work page 2024

  2. [1]

    Tailor made: the art of therapeutic mRNA design,

    M. Metkar, C. S. Pepin, and M. J. Moore, “Tailor made: the art of therapeutic mRNA design,” Nature Reviews Drug Discovery , vol. 23, no. 1, pp. 67–83, 2024

    work page 2024

  3. [2]

    5 ′-utr g-quadruplex structures acting as translational repressors,

    J.-D. Beaudoin and J.-P. Perreault, “5 ′-utr g-quadruplex structures acting as translational repressors,” Nucleic acids research , vol. 38, no. 20, pp. 7022–7036, 2010

    work page 2010

  4. [3]

    mrna structure regulates protein expression through changes in functional half- life,

    D. M. Mauger, B. J. Cabral, V . Presnyak, S. V . Su, D. W. Reid, B. Good- man, K. Link, N. Khatwani, J. Reynders, M. J. Moore et al. , “mrna structure regulates protein expression through changes in functional half- life,” Proceedings of the National Academy of Sciences , vol. 116, no. 48, pp. 24 075–24 083, 2019

    work page 2019

  5. [4]

    Rna strand: the rna secondary structure and statistical analysis database,

    M. Andronescu, V . Bereg, H. H. Hoos, and A. Condon, “Rna strand: the rna secondary structure and statistical analysis database,” BMC bioinformatics, vol. 9, pp. 1–10, 2008

    work page 2008

  6. [5]

    Combinatorial optimization of mrna structure, stability, and translation for rna-based therapeutics,

    K. Leppek, G. W. Byeon, W. Kladwang, H. K. Wayment-Steele, C. H. Kerr, A. F. Xu, D. S. Kim, V . V . Topkar, C. Choe, D. Rothschild et al. , “Combinatorial optimization of mrna structure, stability, and translation for rna-based therapeutics,” Nature communications , vol. 13, no. 1, p. 1536, 2022

    work page 2022

  7. [6]

    Codonbert large language model for mrna vaccines,

    S. Li, S. Moayedpour, R. Li, M. Bailey, S. Riahi, L. Kogler-Anele, M. Miladi, J. Miner, F. Pertuy, D. Zheng et al. , “Codonbert large language model for mrna vaccines,” Genome research, vol. 34, no. 7, pp. 1027–1035, 2024

    work page 2024

  8. [7]

    mrnaid, an open- source platform for therapeutic mrna design and optimization strategies,

    N. V ostrosablin, S. Lim, P. Gopal, K. Brazdilova, S. Parajuli, X. Wei, A. Gromek, D. Prihoda, M. Spale, A. Muzdalo et al., “mrnaid, an open- source platform for therapeutic mrna design and optimization strategies,” NAR Genomics and Bioinformatics , vol. 6, no. 1, p. lqae028, 2024

    work page 2024

Show all 25 references
  1. [8]

    Theoretical basis for stabilizing messenger rna through secondary structure design,

    H. K. Wayment-Steele, D. S. Kim, C. A. Choe, J. J. Nicol, R. Wellington-Oguri, A. M. Watkins, R. A. Parra Sperberg, P.-S. Huang, E. Participants, and R. Das, “Theoretical basis for stabilizing messenger rna through secondary structure design,” Nucleic acids research, vol. 49, ...

    2021

  2. [9]

    Algorithm for optimized mrna design improves stability and immunogenicity,

    H. Zhang, L. Zhang, A. Lin, C. Xu, Z. Li, K. Liu, B. Liu, X. Ma, F. Zhao, H. Jiang et al., “Algorithm for optimized mrna design improves stability and immunogenicity,” Nature, vol. 621, no. 7978, pp. 396–403, 2023

    2023

  3. [10]

    Derna enables pareto optimal rna design,

    X. Gu, Y . Qi, and M. El-Kebir, “Derna enables pareto optimal rna design,” Journal of Computational Biology , vol. 31, no. 3, pp. 179–196, 2024

    2024

  4. [11]

    mrna folding algorithms for structure and codon optimization,

    M. Ward, M. Richardson, and M. Metkar, “mrna folding algorithms for structure and codon optimization,” arXiv preprint arXiv:2503.19273, 2025

    2025 arXiv

  5. [12]

    A resource-efficient varia- tional quantum algorithm for mrna codon optimization,

    H. Zhang, A. Sarkar, and K. Bertels, “A resource-efficient varia- tional quantum algorithm for mrna codon optimization,” arXiv preprint arXiv:2404.14858, 2024

    2024 arXiv

  6. [13]

    Quantum-classical hybrid approach for codon optimization and its practical applications,

    Y . K. Chung, D. Lee, J. Lee, J. Kim, D. K. Park, and J. Huh, “Quantum-classical hybrid approach for codon optimization and its practical applications,” bioRxiv, pp. 2024–06, 2024

    2024

  7. [14]

    The codon adaptation index-a measure of directional synonymous codon usage bias, and its potential applications,

    P. M. Sharp and W.-H. Li, “The codon adaptation index-a measure of directional synonymous codon usage bias, and its potential applications,” Nucleic acids research , vol. 15, no. 3, pp. 1281–1295, 1987

    1987

  8. [15]

    Estimation of secondary structure in ribonucleic acids,

    I. Tinoco, O. C. Uhlenbeck, and M. D. Levine, “Estimation of secondary structure in ribonucleic acids,” Nature, vol. 230, no. 5293, pp. 362–367, 1971

    1971

  9. [16]

    mRNA codon optimization with quantum computers,

    W. R. Fox DM, Branson KM, “mRNA codon optimization with quantum computers,” PLoS One , vol. 16, 2021. doi: 10.1371/jour- nal.pone.0259101

    2021 doi

  10. [17]

    Synonymous but not the same: the causes and consequences of codon bias,

    J. B. Plotkin and G. Kudla, “Synonymous but not the same: the causes and consequences of codon bias,” Nature Reviews Genetics , vol. 12, no. 1, pp. 32–42, 2011

    2011

  11. [18]

    Codon usage tabulated from the inter- national dna sequence databases: status for the year 2000,

    G. T. Nakamura Y . and I. T., “Codon usage tabulated from the inter- national dna sequence databases: status for the year 2000,” Nucl. Acids Res., vol. 28(1):292, 2000

    2000

  12. [20]

    Gusfield, Integer Linear Programming in Computational and Systems Biology: An Entry-Level Text and Course

    D. Gusfield, Integer Linear Programming in Computational and Systems Biology: An Entry-Level Text and Course . Cambridge University Press, 2019

    2019

  13. [21]

    License information

    IBM ILOG CPLEX. License information. Available from IBM: https: //www.ibm.com/products/ilog-cplex-optimization-studio

  14. [22]

    NNDB: the nearest neighbor parameter database for predicting stability of nucleic acid secondary structure,

    D. H. Turner and D. H. Mathews, “NNDB: the nearest neighbor parameter database for predicting stability of nucleic acid secondary structure,” Nucleic Acids Research, vol. 38, no. suppl 1, p. D280– D282, 2009

    2009

  15. [23]

    A simplex method for function minimiza- tion,

    J. A. Nelder and R. Mead, “A simplex method for function minimiza- tion,” The computer journal , vol. 7, no. 4, pp. 308–313, 1965

    1965

  16. [24]

    Improving variational quantum optimization using cvar,

    P. K. Barkoutsos, G. Nannicini, A. Robert, I. Tavernelli, and S. Woerner, “Improving variational quantum optimization using cvar,” Quantum, vol. 4, p. 256, 2020

    2020

  17. [25]

    Viennarna package 2.0,

    R. Lorenz, S. H. Bernhart, C. H ¨oner zu Siederdissen, H. Tafer, C. Flamm, P. F. Stadler, and I. L. Hofacker, “Viennarna package 2.0,” Algorithms for Molecular Biology , vol. 6:1 26, 2011. doi: 10.1186/1748- 7188-6-26

    2011 doi

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