Pith. sign in

REVIEW 4 major objections 3 minor 36 references

Resource-Efficient Quantum Algorithm for Protein Folding

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper's central claim is that the folding problem reduces to finding the ground state of an O(N^4)-term qubit Hamiltonian, and that a variational quantum-classical optimizer can find that state for short peptides on NISQ hardware.

desk verdict Genuinely new encoding and a real hardware demo, but the central reduction to a ground-state problem has a q=0 loophole that breaks self-avoidance, so it needs heavy revision or reframing. read the letter →

arxiv 1908.02163 v1 pith:7S5QN2BK submitted 2019-08-06 quant-ph q-bio.BM

Anton Robert , Panagiotis Kl. Barkoutsos , Stefan Woerner , Ivano Tavernelli This is my paper · ORCID
classification quant-phq-bio.BM MSC 81P68
keywords proteinfoldingquantumHamiltoniantetrahedrallatticevariationaleigensolverCVaR-VQENISQcoarse-grainedmodelgeneticalgorithm
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

This paper tries to establish that protein folding on a tetrahedral lattice can be captured by a qubit Hamiltonian whose ground state is the minimum-energy fold, at a resource cost of O($N^{2}$) qubits and O($N^{4}$) Hamiltonian terms for N monomers. That cost matters because earlier lattice folding Hamiltonians grew much faster, sometimes exponentially, putting them out of reach of near-term quantum devices. The authors pair this Hamiltonian with a hybrid optimizer, CVaR-VQE steered by a genetic algorithm, and report two demonstrations: a noisy simulation of the 10-amino-acid peptide Angiotensin on 22 qubits, and a real-hardware run that folds a 7-amino-acid peptide on 9 qubits, which they describe as the largest variational folding calculation on a NISQ device to date. If the construction holds, it offers a concrete route for using quantum processors to optimize a classical NP-hard problem, not just to simulate quantum systems.

What carries the argument

The machinery is the qubit Hamiltonian $H(q) = H_{gc} + H_{ch} + H_{in}$, defined over two registers. Configuration qubits encode the sequence of turns that grow the chain on the tetrahedral lattice; interaction qubits $q_{i,j}^{(l)}$ mark whether beads $i$ and $j$ form an $l$-th-nearest-neighbour contact. The interaction term $q_{i,j}^{(l)}(\epsilon_{ij}^{(l)} + \lambda(d(i,j)-l))$ gives the contact energy exactly when the distance equals $l$ and the contact qubit is 1, and otherwise applies a large penalty, so the ground state is pushed toward contact patterns with low total energy. Growth and chirality constraints are added as penalties, and the optimisation is done by CVaR-VQE, which minimises the tail of the energy distribution, with a differential-evolution genetic algorithm updating the circuit parameters.

What would settle it

Enumerate all tetrahedral-lattice conformations of a short peptide, compute its exact minimal-energy self-avoiding fold, and compare it with the ground state of $H(q)$; any low-energy state containing a self-intersection far from a contact, or a repulsive contact with its interaction qubit set to zero, would show the claimed equivalence does not hold.

Watch

Extended reading notes

Core claim

The paper's central claim is that the solution to the folding problem is the ground state of the Hamiltonian $H(q)$. The Hamiltonian is built from configuration qubits that encode the polymer's turns on a tetrahedral lattice and interaction qubits that certify pairwise contacts; energy is assigned only when a contact qubit is active and the geometric distance matches, with penalties for mismatches. The authors claim this construction scales as $\mathcal{O}(N^2)$ qubits and $\mathcal{O}(N^4)$ Pauli terms with locality independent of $N$, and they demonstrate it by folding a 7-amino-acid neuropeptide on 9 qubits of a 20-qubit superconducting processor, calling it the largest folding calculation on a NISQ device using a variational algorithm. They also simulate the folding of Angiotensin (10 amino acids) on 22 qubits under a realistic noise model.

Load-bearing premise

The load-bearing premise is that the Hamiltonian's ground state is the minimum-energy self-avoiding fold, but the penalty terms only prevent overlaps that occur near an interaction pair, and repulsive contacts can be ignored by setting the corresponding interaction qubit to zero, so self-intersections elsewhere are not truly forbidden.

Editorial extensions

If this is right

  • The $\mathcal{O}(N^4)$ term count and $\mathcal{O}(N^2)$ qubit count mean lattice protein folding avoids the exponential resource growth of earlier Hamiltonian encodings, provided the construction generalises beyond the demonstrated peptide sizes.
  • On real hardware the 7-amino-acid peptide folded on 9 qubits with the average ground-state probability over the population exceeding 20% and the best individual reaching 33%, which the authors present as evidence that noisy devices can run the algorithm.
  • In noisy simulations the 10-amino-acid Angiotensin system on 22 qubits converged so that at 1024 measurements 100% of the population produced low-energy conformations, and secondary-structure elements such as an alpha-helix and a beta-sheet could be selected by tuning the contact map.
  • Because the contact energies can be taken from pre-existing pairwise interaction tables, the same Hamiltonian can be re-parameterised for different amino-acid sequences without changing the qubit layout.

Reading between the lines

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

  • The paper does not test whether the ground state of $H(q)$ coincides with the true self-avoiding minimum for arbitrary sequences, because overlaps are only penalised in the vicinity of an interaction pair; an exact enumeration for short chains would settle how often unpenalised self-intersections appear in low-energy states.
  • The same CVaR-VQE-plus-genetic-optimizer recipe applies to any diagonal Hamiltonian, so the demonstrated convergence suggests a generic method for classical cost-function optimisation on NISQ hardware, though the paper only shows protein folding.
  • The sparser 4-qubit-per-turn encoding is more faithful to the lattice but uses more qubits; the denser 2-qubit-per-turn encoding trades locality (5-local terms) for fewer qubits, and a systematic comparison of the two encodings on the same peptide would show which is more noise-resilient.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper introduces a coarse-grained quantum algorithm for protein folding on a tetrahedral lattice. The model encodes polymer turns in configuration qubits and pairwise contacts in a separate interaction-qubit register, and defines a Hamiltonian H(q) = H_gc + H_ch + H_in with O(N^4) Pauli terms and O(N^2) qubits. The authors combine CVaR-VQE with a differential-evolution optimizer and report (i) a noisy simulation of the 10-amino-acid Angiotensin peptide on 22 qubits and (ii) a hardware experiment folding a 7-amino-acid neuropeptide on 9 qubits of a 20-qubit IBM Q device. The central claim is that the minimum-energy self-avoiding lattice conformation is obtained as the ground state of H(q).

Significance. If the Hamiltonian construction were sound, the O(N^4) scaling, the quadratic qubit count, and the 9-qubit hardware demonstration would be a useful step toward NISQ-era protein-folding experiments. The use of Miyazawa-Jernigan contact energies for the Angiotensin simulation is a positive feature, as is the validation on real hardware of the CVaR-VQE/genetic-optimizer workflow. However, the central mapping from folding problem to ground state is not established: the Hamiltonian does not enforce global self-avoidance, and the interaction qubits allow the optimizer to remove all overlap penalties and repulsive terms. The secondary-structure demonstration in the SI is also constructed in a circular way. These issues affect the core scientific claim, not merely the presentation.

major comments (4)
  1. [Main text, 'The Hamiltonian' and 'The interaction energy terms'; SI Eq. (SI-30)] The central equivalence between the ground state of H(q) and the minimum-energy self-avoiding fold is not established. The overlap penalties appear only inside h^{(1)}_{ij} in Eq. (SI-30), multiplied by q^{(1)}_{ij}, and the main text explicitly states that 'we only prevent overlaps that occur in the vicinity of an interaction pair.' Since the interaction qubits are variational degrees of freedom, the optimizer can set every q^{(l)}_{ij}=0. In that case H(q) reduces to H_gc + H_ch, which penalizes only immediate back-tracking and chirality (Eqs. SI-17 and SI-26) and does not prevent long-range self-intersection. A self-intersecting configuration with no selected contacts is therefore energetically degenerate with a valid no-contact fold, and for sequences whose optimized q register is all-zero the ground state is not a valid protein conformation. This invalidates the claim in the folding-algorithm section that 'the solution to the folding problem is the ground state of H(q)'.
  2. [Main text, 'Folding algorithm' section and Fig. 2(b)] The assertion that the ground state 'lies in the 2^{N_cf} dimensional space of the configuration qubits' is not correct as stated: H(q) acts on the joint register (q_cf, q_in), and the minimization is performed over both registers. The energy labels in Fig. 2(b) are reported in terms of contact-qubit strings rather than full configuration strings, so the low-energy states are not shown to correspond to self-avoiding conformations of the polymer. A valid ground-state computation must optimize over q_in and then verify that the resulting configuration is self-avoiding; the manuscript does neither.
  3. [SI, Section II, Fig. S2] The secondary-structure demonstration is circular as presented. The contact maps in Fig. S2 are explicitly designed to stabilize an alpha-helix (upper triangle) or an antiparallel beta-sheet (lower triangle), and the simulations then reproduce those expected structures. This does not provide evidence that the model predicts secondary structure from sequence; it only shows that the optimizer can find the designed minimum of a hand-built cost function. This section should be reframed as a consistency check rather than as a reproduction of secondary structure.
  4. [Main text, 'The interaction energy terms' and Discussion] The model cannot represent repulsive interactions. The term q^{(l)}_{ij}(epsilon^{(l)}_{ij}+lambda(d(i,j)-l)) contributes only when the interaction qubit is 1. For a repulsive interaction with epsilon > 0, the optimizer can set q=0 and remove the energy penalty entirely, so configurations in which repulsive pairs are in contact are never penalized. The Discussion's statement that the model can account for Lennard-Jones-like interactions is therefore unsupported by the Hamiltonian as written.
minor comments (3)
  1. [Introduction and Discussion, spelling] There are typos in the text, including 'curently' in the Introduction and 'Hamltonain'/'Hamltonian' in the Discussion; these should be corrected.
  2. [SI Eq. (SI-30) and Materials and Methods] The penalty parameters lambda_1, lambda_2, lambda_3, lambda_5 and the chirality/backtracking penalties lambda_back, lambda_chirality are not given numerical values for the reported simulations; please provide the values used and state how they were chosen to dominate the contact energies.
  3. [Main text, 'Applications' and Fig. 3(e)] The text reports max_p P0(p) = 42.2% for the Angiotensin simulation and later says max_p P0(p) peaks at 33% for the 7-amino-acid hardware run; clarify in the figure caption and text which probability is being reported for each system.

Circularity Check

1 steps flagged · score 4.0 of 10

Secondary-structure 'reproduction' is input-to-output via a tuned contact map, while the main folding result relies on external Miyazawa-Jernigan energies; partial circularity only.

  1. fitted input called prediction [Applications (main text) and SI section 'II. CONTACT MAP', Fig. S2]
    "By tuning the interaction matrix (see Fig.S2 in SI), we can foster the formation of secondary structural elements. ... Contact maps for the stabilization of two different secondary structure elements: β-sheet (left) versus α-helix (right). Simulations performed with these contact maps reproduce the correct minimal energy structures."

    The contact map in Fig. S2 is the input interaction matrix, and the caption says it is explicitly designed to stabilize an α-helix (contacts parallel to the diagonal) and an anti-parallel β-sheet (contacts along the counter-diagonal). The Hamiltonian assigns favorable energies to exactly those contacts, so the optimizer returning those structures is forced by construction. Presenting this as 'reproducing' secondary-structure elements is input-to-output, not an independent prediction. The paper itself concedes this tunability: the pairwise interaction energies 'can be arbitrarily defined to reproduce a fold of interest.' Thus the secondary-structure claim reduces to a fitted input renamed as a reproduced prediction.

full rationale

The central folding construction is not circular: the interaction energies for the Angiotensin simulation come from the pre-existing Miyazawa-Jernigan contact-energy table (ref 27), the qubit encoding and O(N^4) scaling are derived from the tetrahedral-lattice geometry, and the 7-amino-acid IBM Q experiment is an external hardware demonstration. The CVaR-VQE optimizer is the authors' own prior method (ref 30), but it is an optimization subroutine and does not by itself determine which fold is found. The one genuine input-to-output episode is the secondary-structure demonstration: the SI contact map is deliberately tuned to stabilize an α-helix or β-sheet, and the simulation then 'reproduces' those exact structures. This is a supporting demonstration rather than the central claim, and the main fold predictions still rest on external MJ energies, so the circularity is partial rather than pervasive. Score 4 reflects that one fitted input is presented as a reproduction, while the core algorithm retains independent content.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The central claim depends on a set of hand-chosen penalty weights, an interaction cutoff, a fitted scaling curve, and optimization hyperparameters. The modeling rests on domain assumptions about lattice coarse-graining and transferability of MJ energies. The invented interaction-qubit register is a computational device with no independent falsifiable handle.

free parameters (6)
  • lambda_back (backtracking penalty) = large positive, not specified
    Used in Hgc (SI Eq. SI-17) to forbid the chain choosing the same axis twice in a row; the exact value is not reported and the central claim depends on it being large enough to dominate contact energies.
  • lambda_chirality = large positive, not specified
    Used in Hch (SI Eq. SI-26); enforces side-chain chirality.
  • lambda1, lambda2 (and lambda3, lambda5 for l=2) = large positive, constrained by inequalities such as lambda1 > 6(j-i+1)lambda2 + epsilon, values not reported
    Distance-mismatch and overlap penalties in Hin (main text, Eqs. SI-29/30 and SI-34/35); the ground-state equivalence relies on these values.
  • scaling fit parameters (a,b) = (0.15, 1.49)
    Fit to N_t versus N in Fig. 1(e) to illustrate O(N^4) scaling; a fitted curve rather than a derived count.
  • l-NN cutoff l_max = l=1 for the two demonstrations; general model unspecified
    Number of interaction qubits grows exponentially with l, so the O(N^4) resource claim holds only for fixed cutoff; experiments neglect l>1.
  • CVaR alpha, VQE depth m, DE population P=5mn = alpha=1% (128 shots), 0.1% (1024 shots); m=2; P=5mn
    Optimization hyperparameters chosen by hand from literature; they influence convergence and are needed to reproduce the experiments.
assumptions (5)
  • standard math Tetrahedral lattice distance map d(i,j)=sum_a Delta n_a^2 is a bijection with Euclidean lattice distances (SI Eq. SI-14).
    Used to identify contacts at distance l from turn qubit configurations.
  • domain assumption A protein can be modeled as a self-avoiding walk of two-centered beads on a tetrahedral lattice with MJ contact energies.
    The whole folding model rests on this coarse-graining; the paper provides no validation against experimental structures.
  • domain assumption Miyazawa-Jernigan statistical contact energies transfer to lattice nearest-neighbor contacts.
    The pair interaction matrix for Angiotensin is taken from Table 3 of ref 27.
  • ad hoc to paper Setting q(l)=1 only for attractive contacts yields the correct ground state; repulsive interactions can be ignored.
    The q-qubit construction q(epsilon+lambda(d-l)) lets the optimizer set q=0 to avoid repulsive energies, so repulsion is not represented (main text interaction energy terms).
  • ad hoc to paper Unpenalized overlaps away from interaction pairs do not affect the low-energy spectrum.
    The paper only prevents overlaps near contacts ('we only prevent overlaps that occur in the vicinity of an interaction pair'), so the equivalence between ground state and valid fold assumes other overlaps are harmless.
invented entities (1)
  • Interaction qubits q^{(l)}_{ij}
    purpose: Register of qubits marking whether beads i and j form an l-th nearest-neighbor contact, enabling a local Hamiltonian term q(epsilon+lambda(d-l)).
    Introduced by this paper to keep Hamiltonian locality independent of N; they are computational artifacts with no observable outside the model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Resource-Efficient Quantum Algorithm for Protein Folding." pith.science (2026). https://pith.science/paper/7S5QN2BK

@misc{pith2026190802163,
  author       = {Pith},
  title        = {Pith review of: Resource-Efficient Quantum Algorithm for Protein Folding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7S5QN2BK}},
  note         = {Machine review of arXiv:1908.02163}
}
abstract

Predicting the three-dimensional (3D) structure of a protein from its primary sequence of amino acids is known as the protein folding (PF) problem. Due to the central role of proteins' 3D structures in chemistry, biology and medicine applications (e.g., in drug discovery) this subject has been intensively studied for over half a century. Although classical algorithms provide practical solutions, sampling the conformation space of small proteins, they cannot tackle the intrinsic NP-hard complexity of the problem, even reduced to its simplest Hydrophobic-Polar model. While fault-tolerant quantum computers are still beyond reach for state-of-the-art quantum technologies, there is evidence that quantum algorithms can be successfully used on Noisy Intermediate-Scale Quantum (NISQ) computers to accelerate energy optimization in frustrated systems. In this work, we present a model Hamiltonian with $\mathcal{O}(N^4)$ scaling and a corresponding quantum variational algorithm for the folding of a polymer chain with $N$ monomers on a tetrahedral lattice. The model reflects many physico-chemical properties of the protein, reducing the gap between coarse-grained representations and mere lattice models. We use a robust and versatile optimisation scheme, bringing together variational quantum algorithms specifically adapted to classical cost functions and evolutionary strategies (genetic algorithms), to simulate the folding of the 10 amino acid Angiotensin peptide on 22 qubits. The same method is also successfully applied to the study of the folding of a 7 amino acid neuropeptide using 9 qubits on an IBM Q 20-qubit quantum computer. Bringing together recent advances in building gate-based quantum computers with noise-tolerant hybrid quantum-classical algorithms, this work paves the way towards accessible and relevant scientific experiments on real quantum processors.

Figures

Figures reproduced from arXiv: 1908.02163 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 16 canonical work pages

  1. [1]

    Levitt \ and\ author A

    author author M. Levitt \ and\ author A. Warshel ,\ @noop journal journal Nature \ volume 253 ,\ pages 694 ( year 1975 ) NoStop

    work page 1975

  2. [2]

    Zwanzig , author A

    author author R. Zwanzig , author A. Szabo , \ and\ author B. Bagchi ,\ 10.1073/pnas.89.1.20 journal journal Proceedings of the National Academy of Sciences \ volume 89 ,\ pages 20 ( year 1992 ) NoStop

    doi:10.1073/pnas.89.1.20 1992

  3. [3]

    author author D. A. \ Hinds \ and\ author M. Levitt ,\ 10.1073/pnas.89.7.2536 journal journal Proceedings of the National Academy of Sciences \ volume 89 ,\ pages 2536 ( year 1992 ) NoStop

    work page doi:10.1073/pnas.89.7.2536 1992

  4. [4]

    author author E. I. \ Shakhnovich ,\ 10.1103/PhysRevLett.74.2618 journal journal Physical Review Letters \ volume 74 ,\ pages 2618 ( year 1995 ) NoStop

    work page doi:10.1103/physrevlett.74.2618 1995

  5. [5]

    author author J. N. \ Onuchic \ and\ author P. G. \ Wolynes ,\ 10.1016/j.sbi.2004.01.009 journal journal Current Opinion in Structural Biology \ volume 14 ,\ pages 70 ( year 2004 ) NoStop

    work page doi:10.1016/j.sbi.2004.01.009 2004

  6. [6]

    Lindorff-Larsen , author S

    author author K. Lindorff-Larsen , author S. Piana , author R. O. \ Dror , \ and\ author D. E. \ Shaw ,\ 10.1126/science.1208351 journal journal Science \ volume 334 ,\ pages 517 ( year 2011 ) NoStop

    doi:10.1126/science.1208351 2011

  7. [7]

    Hyeon \ and\ author D

    author author C. Hyeon \ and\ author D. Thirumalai ,\ 10.1038/ncomms1481 journal journal Nature Communications \ volume 2 ,\ pages 487 ( year 2011 ) NoStop

    work page doi:10.1038/ncomms1481 2011

  8. [8]

    Unger \ and\ author J

    author author R. Unger \ and\ author J. Moult ,\ 10.1016/S0092-8240(05)80169-7 journal journal Bulletin of Mathematical Biology \ volume 55 ,\ pages 1183 ( year 1993 ) NoStop

    work page doi:10.1016/s0092-8240(05)80169-7 1993

Show all 36 references
  1. [9]

    Berger \ and\ author T

    author author B. Berger \ and\ author T. Leighton ,\ 10.1089/cmb.1998.5.27 journal journal Journal of Computational Biology \ volume 5 ,\ pages 27 ( year 1998 ) NoStop

    1998 doi

  2. [10]

    Moll , author P

    author author N. Moll , author P. Barkoutsos , author L. S. \ Bishop , author J. M. \ Chow , author A. Cross , author D. J. \ Egger , author S. Filipp , author A. Fuhrer , author J. M. \ Gambetta , author M. Ganzhorn , author A. Kandala , author A. Mezzacapo , author P. Müller...

    doi

  3. [11]

    Preskill ,\ http://arxiv.org/abs/1801.00862 journal journal arXiv:1801.00862 [cond-mat, physics:quant-ph] \ ( year 2018 ) ,\ note arXiv: 1801.00862 NoStop

    author author J. Preskill ,\ http://arxiv.org/abs/1801.00862 journal journal arXiv:1801.00862 [cond-mat, physics:quant-ph] \ ( year 2018 ) ,\ note arXiv: 1801.00862 NoStop

    2018 arXiv

  4. [12]

    Mandr\`a \ and\ author H

    author author S. Mandr\`a \ and\ author H. G. \ Katzgraber ,\ 10.1088/2058-9565/aac8b2 journal journal Quantum Science and Technology \ volume 3 ,\ pages 04LT01 ( year 2018 ) ,\ note arXiv: 1711.01368 NoStop

    2018 arXiv

  5. [14]

    King , author S

    author author J. King , author S. Yarkoni , author J. Raymond , author I. Ozfidan , author A. D. \ King , author M. M. \ Nevisi , author J. P. \ Hilton , \ and\ author C. C. \ McGeoch ,\ 10.1103/PhysRevX.6.031015 journal journal arXiv:1701.04579 [quant-ph] \ volume 6 ( year 20...

    arXiv

  6. [15]

    Streif \ and\ author M

    author author M. Streif \ and\ author M. Leib ,\ http://arxiv.org/abs/1901.01903 journal journal arXiv:1901.01903 [quant-ph] \ ( year 2019 ) ,\ note arXiv: 1901.01903 NoStop

    1901 arXiv

  7. [16]

    Levinthal ,\ 10.1051/jcp/1968650044 journal journal Journal de Chimie Physique \ ,\ pages 44 ( year 1968 ) NoStop

    author author C. Levinthal ,\ 10.1051/jcp/1968650044 journal journal Journal de Chimie Physique \ ,\ pages 44 ( year 1968 ) NoStop

    1968

  8. [17]

    author author J. N. \ Onuchic , author P. G. \ Wolynes , author Z. Luthey-Schulten , \ and\ author N. D. \ Socci ,\ 10.1073/pnas.92.8.3626 journal journal Proceedings of the National Academy of Sciences \ volume 92 ,\ pages 3626 ( year 1995 ) NoStop

    doi

  9. [18]

    Piana , author J

    author author S. Piana , author J. L. \ Klepeis , \ and\ author D. E. \ Shaw ,\ 10.1016/j.sbi.2013.12.006 journal journal Current Opinion in Structural Biology \ volume 24 ,\ pages 98 ( year 2014 ) NoStop

    2013 doi

  10. [19]

    Duan ,\ 10.1126/science.282.5389.740 journal journal Science \ volume 282 ,\ pages 740 ( year 1998 ) NoStop

    author author Y. Duan ,\ 10.1126/science.282.5389.740 journal journal Science \ volume 282 ,\ pages 740 ( year 1998 ) NoStop

    1998 doi

  11. [20]

    Babbush , author A

    author author R. Babbush , author A. Perdomo-Ortiz , author B. O'Gorman , author W. Macready , \ and\ author A. Aspuru-Guzik ,\ @noop journal journal arXiv preprint arXiv:1211.3422 \ ( year 2012 ) NoStop

    2012 arXiv

  12. [21]

    Perdomo Ortiz , author C

    author author A. Perdomo Ortiz , author C. Truncik , author I. Tubert-Brohman , author G. Rose , \ and\ author A. Aspuru-Guzik ,\ @noop journal journal Physical Review A \ volume 78 ( year 2008 ) NoStop

    2008

  13. [22]

    Perdomo-Ortiz , author N

    author author A. Perdomo-Ortiz , author N. Dickson , author M. Drew-Brook , author G. Rose , \ and\ author A. Aspuru-Guzik ,\ 10.1038/srep00571 journal journal Scientific Reports \ volume 2 ( year 2012 ),\ 10.1038/srep00571 NoStop

    2012 doi

  14. [23]

    Babej , author C

    author author T. Babej , author C. Ing , \ and\ author M. Fingerhuth ,\ http://arxiv.org/abs/1811.00713 journal journal arXiv:1811.00713 [quant-ph] \ ( year 2018 ) ,\ note 00002 arXiv: 1811.00713 NoStop

    2018 arXiv

  15. [24]

    Cai , author W

    author author J. Cai , author W. G. \ Macready , \ and\ author A. Roy ,\ http://arxiv.org/abs/1406.2741 journal journal arXiv:1406.2741 [quant-ph] \ ( year 2014 ) ,\ note arXiv: 1406.2741 NoStop

    2014 arXiv

  16. [25]

    Farhi \ and\ author A

    author author E. Farhi \ and\ author A. W. \ Harrow ,\ http://arxiv.org/abs/1602.07674 journal journal arXiv:1602.07674 [quant-ph] \ ( year 2016 ) NoStop

    2016 arXiv

  17. [26]

    Fingerhuth , author T

    author author M. Fingerhuth , author T. Babej , \ and\ author C. Ing ,\ http://arxiv.org/abs/1810.13411 journal journal arXiv:1810.13411 [quant-ph] \ ( year 2018 ) ,\ note 00002 arXiv: 1810.13411 NoStop

    2018 arXiv

  18. [27]

    Miyazawa \ and\ author R

    author author S. Miyazawa \ and\ author R. L. \ Jernigan ,\ @noop journal journal Journal of molecular biology \ volume 256 ,\ pages 623 ( year 1996 ) NoStop

    1996

  19. [28]

    Peruzzo , author J

    author author A. Peruzzo , author J. McClean , author P. Shadbolt , author M.-H. \ Yung , author X.-Q. \ Zhou , author P. J. \ Love , author A. Aspuru-Guzik , \ and\ author J. L. \ O'Brien ,\ 10.1038/ncomms5213 journal journal Nat Commun \ volume 5 ( year 2014 ),\ 10.1038/ncom...

    doi

  20. [29]

    author author J. R. \ McClean , author J. Romero , author R. Babbush , \ and\ author A. Aspuru-Guzik ,\ 10.1088/1367-2630/18/2/023023 journal journal New J. Phys. \ volume 18 ,\ pages 023023 ( year 2016 ) NoStop

    2016 doi

  21. [30]

    author author P. K. \ Barkoutsos , author G. Nannicini , author A. Robert , author I. Tavernelli , \ and\ author S. Woerner ,\ http://arxiv.org/abs/1907.04769 journal journal arXiv:1907.04769 [quant-ph] \ ( year 2019 ) ,\ note 00000 arXiv: 1907.04769 NoStop

    1907 arXiv

  22. [31]

    Storn \ and\ author K

    author author R. Storn \ and\ author K. Price ,\ 10.1023/A:1008202821328 journal journal Journal of Global Optimization \ volume 11 ,\ pages 341 ( year 1997 ) NoStop

    1997 doi

  23. [32]

    Hadfield , author Z

    author author S. Hadfield , author Z. Wang , author B. O'Gorman , author E. G. \ Rieffel , author D. Venturelli , \ and\ author R. Biswas ,\ 10.3390/a12020034 journal journal Algorithms \ volume 12 ,\ pages 34 ( year 2019 ) ,\ note arXiv: 1709.03489 NoStop

    2019 arXiv

  24. [33]

    Aleksandrowicz , author T

    author author G. Aleksandrowicz , author T. Alexander , author P. Barkoutsos , author L. Bello , author Y. Ben-Haim , author D. Bucher , author F. J. \ Cabrera-Hern \'a ndez , author J. Carballo-Franquis , author A. Chen , author C.-F. \ Chen , author J. M. \ Chow , author A. ...

  25. [34]

    Das , author S

    author author S. Das , author S. S. \ Mullick , \ and\ author P. Suganthan ,\ 10.1016/j.swevo.2016.01.004 journal journal Swarm and Evolutionary Computation \ volume 27 ,\ pages 1 ( year 2016 ) NoStop

    2016 doi

  26. [35]

    Perdomo , author C

    author author A. Perdomo , author C. Truncik , author I. Tubert-Brohman , author G. Rose , \ and\ author A. Aspuru-Guzik ,\ @noop journal journal Physical Review A \ volume 78 ( year 2008 ) NoStop

    2008

  27. [36]

    Kempe , author A

    author author J. Kempe , author A. Kitaev , \ and\ author O. Regev ,\ @noop journal journal SIAM J. Comput. \ volume 35 ,\ pages 30 ( year 2006 ) NoStop

    2006

  28. [37]

    author author P. K. \ Barkoutsos , author J. F. \ Gonthier , author I. Sokolov , author N. Moll , author G. Salis , author A. Fuhrer , author M. Ganzhorn , author D. J. \ Egger , author M. Troyer , author A. Mezzacapo , author S. Filipp , \ and\ author I. Tavernelli ,\ 10.1103...

    doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.