Pith. sign in

REVIEW 3 major objections 3 minor 122 references

Bridging Quantum Chemistry and MaxCut: Classical Performance Guarantees and Quantum Algorithms for the Hartree-Fock Method

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper shows that Hartree-Fock SCF optimization is exactly a sequence of MaxCut problems, so the selected Slater determinant carries a provable energy guarantee at every step.

desk verdict The MaxCut reformulation is real and the numerics are respectable, but the advertised energy performance guarantee is unsupported: the approximation ratio applies to the cut objective, and the affine shift to the energy breaks the multiplicative bound. read the letter →

arxiv 2506.04223 v1 pith:2LCDZYVN submitted 2025-06-04 quant-ph cond-mat.mtrl-sciphysics.chem-phphysics.comp-ph

Alexis Ralli , Tim Weaving , Peter V. Coveney , Peter J. Love This is my paper
classification quant-phcond-mat.mtrl-sciphysics.chem-phphysics.comp-ph MSC 90C2281P6881V55
keywords Hartree-FockMaxCutQUBOsemidefiniteprogrammingself-consistentfieldJordan-WignertransformationquantumoptimizationGoemans-Williamsonalgorithm
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 claims that the Hartree-Fock self-consistent field (SCF) calculation can be rewritten, step by step, as a sequence of quadratic unconstrained spin/binary optimization (QUSO/QUBO) problems, and equivalently as MaxCut graph problems. Because each inner step is exactly a MaxCut instance, the Goemans-Williamson approximation ratio applies to the energy of the Slater determinant selected in that step, which gives a performance guarantee no conventional SCF solver offers. The authors demonstrate QUBO-SCF and MaxCut-SCF on the hydroxide anion and on molecular nitrogen with basis sets requiring up to 220 qubits, and report that the new methods are less prone to internal instabilities and can improve single-reference post-Hartree-Fock results such as CISD. If the claim holds, any QUBO/Ising optimizer, classical or quantum, can serve as the inner engine of Hartree-Fock while keeping a guarantee on the quality of the reference state.

What carries the argument

The load-bearing object is the diagonal-in-Fock-basis Hamiltonian H_D(κ) (Equation 4), whose Jordan-Wigner image contains only identity and two-body Z terms, making it a quadratic unconstrained spin optimization problem by construction. The MaxCut mapping appends a single ancillary spin so every term becomes a two-spin product, converting the minimization into edge-weighted MaxCut on a graph whose positive edge weights scale quadratically with the number of spin-orbitals and whose negative edge weights scale linearly. The Goemans-Williamson semidefinite programming relaxation with hyperplane rounding, extended to negative-weight graphs, then supplies the per-step approximation ratio; the outside loop updates the orbital basis with the standard orbital gradient and Hessian (Algorithm 1) or Fock-matrix diagonalization (Algorithm 2).

What would settle it

Take a small basis-set instance where the symmetry penalties are active, solve the MaxCut semidefinite program to optimality, and test whether the rounded cut ever violates the approximation inequality (F8); finding one counterexample would falsify the per-step performance guarantee as stated.

Watch

Extended reading notes

Core claim

The central discovery is that the difficulty of Hartree-Fock splits cleanly into an outer continuous rotation of the orbital basis and an inner discrete choice of which Slater determinant to occupy, and the inner choice is exactly a two-local diagonal spin problem. For any fixed orbital basis, the diagonal part of the molecular Hamiltonian is already a QUSO after the Jordan-Wigner transform, and adding one ancillary spin turns it into a MaxCut graph with edge weights read off from the one- and two-electron integrals. Consequently, at each SCF iteration one can provably select a Slater determinant whose energy is within a known constant fraction of the best determinant in that basis; the paper derives the relevant approximation bound for graphs with negative edge weights, notes that positive edges dominate because the two-electron integrals are positive semidefinite, and verifies the procedure on OH- and N2 up to 220 qubits.

Load-bearing premise

The central claim depends on the assumption that the alternating inner and outer optimization converges to the desired Hartree-Fock minimum rather than getting stuck in a local minimum; no convergence proof is supplied, and the numerical evidence covers only two molecules.

Editorial extensions

If this is right

  • Every SCF iteration gains a certificate: the chosen Slater determinant's energy is at least a known fraction of the optimal determinant's energy in the current basis, a property absent from conventional Hartree-Fock.
  • QUBO-SCF and MaxCut-SCF avoid many internal instabilities because the reference determinant is re-optimized at each step instead of being fixed, as demonstrated on OH- and stretched N2.
  • The orbitals produced by the new SCF methods can improve single-reference post-Hartree-Fock calculations such as CISD, since the virtual orbitals are effectively optimized during the SCF.
  • The inner optimization is a QUSO/QUBO, so any Ising-machine or quantum optimizer (QAOA, quantum annealing, Grover adaptive search, decoded quantum interferometry) can replace the classical inner solver without changing the outer orbital update.
  • The method remains NP-hard at each step, but the hardness is now accompanied by a quantitative approximation guarantee rather than an uncontrolled local-minimum risk.

Reading between the lines

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

  • If the symmetry-penalty terms preserve the approximation inequality (a point the paper leaves open), MaxCut-SCF would supply the first worst-case energy guarantee for Hartree-Fock in a fixed basis; testing small penalized instances directly would settle this.
  • Because the same diagonal-Hamiltonian structure exists for Kohn-Sham DFT orbitals, the MaxCut machinery likely extends to KS-DFT orbital optimization, with the same per-step guarantees inherited from the graph formulation.
  • The CISD improvements seen in the paper suggest a practical recipe unrelated to the approximation guarantee: use MaxCut-SCF orbitals as the starting point for coupled-cluster or perturbation methods and benchmark on larger molecules.
  • The 220-qubit MaxCut instances generated here are natural test cases for near-term quantum optimizers, since the objective is a sparse MAX-2-XORSAT instance that decoded quantum interferometry is designed to solve.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper reformulates the Hartree-Fock self-consistent field (SCF) optimization as a sequence of Quadratic Unconstrained Spin/Binary Optimization (QUSO/QUBO) problems by taking the diagonal part of the molecular Hamiltonian in a rotating orbital basis. It then maps these QUSO problems to MaxCut using one ancilla qubit and invokes the Goemans-Williamson approximation ratio to claim a per-step performance guarantee on the energy of the selected Slater determinant. The authors test QUBO-SCF and MaxCut-SCF on OH- and N2 in basis sets up to 220 qubits, and outline four hybrid quantum-classical algorithms (GAS-SCF, QAOA-SCF, QA-SCF, DQI-SCF).

Significance. If the energy-fraction guarantee were valid, this would be a significant conceptual advance: it would add an approximation-theoretic guarantee to a standard quantum-chemistry method and connect it to well-studied combinatorial optimization. The exact QUSO formulation, the explicit ancilla MaxCut mapping, and the detailed numerical data are useful and reproducible. However, the advertised energy guarantee is not established, and the practical claims rest on only two molecular studies. The contribution is nonetheless of interest to the quantum-optimization and quantum-chemistry intersection, provided the claims are re-scoped.

major comments (3)
  1. [Appendix E2 (Eqs. E10-E12) and Appendix F (Eq. F8); Abstract; Section V] The Goemans-Williamson guarantee is for the MaxCut objective C(z)=sum_e w_e(1-z_u z_v)/2, not for the QUSO energy E(z). In moving from Eq. (E10) to Eq. (E12) the derivation discards the constant terms -cI/2 and the I/2 inside the second sum, so the computed cut value is C(z) = -E(z) + K with K = O(sum |w_e|) never bounded. Eq. (F8) therefore gives C_approx >= alpha C_opt + (1-alpha) W^-, which implies E_approx <= alpha E_opt + (1-alpha)(K + W^-). Since K is on the electronic-energy scale and W^- can be inflated by penalty terms, the selected determinant's energy is not guaranteed to be any fixed fraction of the optimal energy. The abstract and Section V claim exactly this energy fraction; that claim is unsupported and must be either proved with an explicit bound on K or replaced by a cut-value guarantee.
  2. [Equations (11)-(12), Algorithms 1-2 (Appendix I)] The two-level minimization is solved by alternating an inner determinant search with an outer orbital update, but no convergence proof is given for this alternating procedure. It is not shown that the sequence of iterates converges to a stationary point of Eqs. (11)-(12), nor that the per-step guarantee composes over iterations. The numerical evidence spans only two molecules, so the abstract's claim that the procedure works 'irrespective of the complexity of the optimization landscape' extends beyond what is demonstrated. A formal convergence statement (even to a local optimum) or an explicit caveat is needed.
  3. [Appendix B and Section IV (numerical setup)] The performance guarantee in Eq. (F8) is derived for the unpenalized MaxCut graph, but all numerical experiments add the penalty terms described in Appendix B with lambda chosen as the Hamiltonian 1-norm. No proof is given that this lambda enforces the desired symmetry sectors, and Appendix E2 notes that the MaxCut derivation 'contains no penalization terms' and omits their analysis. As a result, the claimed per-step guarantee does not apply to the numerically implemented objective. The authors should either analyze the effect of penalization on the approximation ratio or explicitly exclude penalized problems from the guarantee.
minor comments (3)
  1. [Equation (E12)] The double sum over ordered pairs (m,n) double-counts graph edges compared with the standard MaxCut convention sum_{i<j}; a sentence clarifying the relation (or converting to m<n) would avoid confusion.
  2. [Section IIID (DQI-SCF)] The mapping in Eqs. (18)-(19) turns the weighted MaxCut objective into an unweighted MAX-2-XORSAT clause set, which is not equivalent to the original optimization; the DQI discussion should state that this is a further approximation or restrict the claim to unweighted instances.
  3. [Section IV] Reporting energies to six decimals is adequate, but the comparison of different codes (Psi4, Gaussian, Orca) should state what convergence thresholds and initial guesses were used, since SAD initial guesses are standard only in PySCF.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QUSO/QUBO coefficients come directly from molecular integrals, and the performance guarantees are imported from external semidefinite programming theory rather than from the paper's own outputs.

full rationale

The derivation chain is self-contained against external benchmarks. Equations (11)-(14) define an exact alternating reformulation of HF-style SCF: for fixed orbital basis R, the diagonal Hamiltonian H_D(R) is written as a QUSO via the Jordan-Wigner map in Equation (7), and the inner minimization over basis states is a genuine combinatorial problem whose coefficients are the molecular integrals; no fitted parameter encodes the answer. The MaxCut mapping in Equations (E10)-(E12) uses one ancilla and preserves the argmax, so the QUSO/QUBO-to-MaxCut equivalence is structurally sound; the only hand-chosen quantity, the penalty strength lambda = 1-norm, is a heuristic that does not encode the solution. The Goemans-Williamson and negative-weight bounds in Appendix F are external theorems, and the numerical benchmarks use independent codes (PySCF, Psi4, Gaussian, Orca). The few self-citations (Refs. [19], [69], [87] include P.J. Love) are background on FCI quantum algorithms, VQE, and fermion-to-qubit mappings, and none is load-bearing for the central claim. The skeptics' affine-shift objection is a correctness concern rather than a circularity: Equation (F8) bounds the MaxCut objective, while Equations (E10)-(E12) show that objective differs from the energy by dropped constants (-cI/2 and -I/2), so the claimed transfer of the 0.878 ratio from cut value to energy is not established by the paper's equations; this missing step does not reduce any output to an input, so it does not raise the circularity score. Similarly, the lack of a convergence proof for the outer SCF loop (Algorithms 1-2) and the two-molecule numerical basis are limitations explicitly acknowledged in the Conclusion, not circular reasoning.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard HF theory, the QUSO-to-MaxCut reduction, and external SDP approximation theorems. The only hand-set parameter is the penalty strength, and the outer-loop convergence is assumed rather than proved.

free parameters (2)
  • penalty strength λ = 1-norm of the Hamiltonian
    Used in the numerical QUBO/QUSO/MaxCut instances to enforce S2=0 and correct electron number. Chosen by hand as the Hamiltonian 1-norm; sensitivity not analyzed.
  • GW hyperplane sample count = not reported
    The Goemans-Williamson rounding is randomized, and the expected approximation ratio depends on sampling; the number of samples used in the numerics is not stated.
assumptions (5)
  • domain assumption Non-relativistic Born-Oppenheimer second-quantized molecular Hamiltonian is the correct model for the studied systems
    Invoked in Section IIA, Equation (1). Standard model in quantum chemistry; not rederived.
  • domain assumption Two-electron integrals satisfy (g_mmnn - g_mnnm) ≥ 0 for real orbitals
    Used in Section IIIB and Appendix E2 to bound the number of positive MaxCut edges; relies on positive-definiteness of the Coulomb operator [26].
  • standard math Goemans-Williamson SDP algorithm achieves 0.878 approximation for nonnegative-weight MaxCut, with the stated generalizations for negative weights
    External theorem used in Section IIIB and Appendix F to support the per-step performance guarantee.
  • ad hoc to paper Penalty strength λ = 1-norm is large enough to enforce the desired symmetry sectors
    Assumed in the numerical study (Section IV); no proof or sensitivity study is provided.
  • domain assumption The alternating optimization (inner QUBO over determinants, outer orbital update) converges to a stationary point and, in practice, to the global HF minimum
    Relied on by Algorithms 1 and 2 (Section IIIA and Appendix I). The paper provides no convergence proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bridging Quantum Chemistry and MaxCut: Classical Performance Guarantees and Quantum Algorithms for the Hartree-Fock Method." pith.science (2026). https://pith.science/paper/2LCDZYVN

@misc{pith2026250604223,
  author       = {Pith},
  title        = {Pith review of: Bridging Quantum Chemistry and MaxCut: Classical Performance Guarantees and Quantum Algorithms for the Hartree-Fock Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LCDZYVN}},
  note         = {Machine review of arXiv:2506.04223}
}
read the original abstract

In quantum chemistry, self-consistent field (SCF) algorithms define a nonlinear optimization problem, with both continuous and discrete components. In this work, we derive Hartree-Fock-inspired SCF algorithms that can be exactly written as a sequence of Quadratic Unconstrained Spin/Binary Optimization problems (QUSO/QUBO). We reformulate the optimization problem as a series of MaxCut graph problems, which can be efficiently solved using semi-definite programming techniques. This procedure provides performance guarantees at each SCF step, irrespective of the complexity of the optimization landscape. We numerically demonstrate the QUBO-SCF and MaxCut-SCF methods by studying the hydroxide anion OH- and molecular Nitrogen N2. The largest problem addressed in this study involves a system comprised of 220 qubits (equivalently, spin-orbitals). Our results show that QUBO-SCF and MaxCut-SCF suffer much less from internal instabilities compared with conventional SCF calculations. Additionally, we show that the new SCF algorithms can enhance single-reference methods, such as configuration interaction. Finally, we explore how quantum algorithms for optimization can be applied to the QUSO problems arising from the Hartree-Fock method. Four distinct hybrid-quantum classical approaches are introduced: GAS-SCF, QAOA-SCF, QA-SCF and DQI-SCF.

Figures

Figures reproduced from arXiv: 2506.04223 by the authors.

Figure 1
Figure 1. FIG. 1: Outline of the new SCF algorithms derived in this work. (a) Algorithm [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Converge of different SCF optimization for OH [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: SCF potential energy surface for the dissociation of N [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: CISD potential energy surface for the dissociation of N [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

122 extracted references · 68 canonical work pages

  1. [1]

    The main idea of this approach is to iteratively apply the Grover search algo- rithm to find the optimum value of an objective func- tion

    GAS-SCF Gilliam, Woerner, and Gonciulea [65] recently demon- strated that quadratic unconstrained binary optimiza- tion (QUBO) problems can be accelerated using Grover Adaptive Search (GAS) [66, 67]. The main idea of this approach is to iteratively apply the Grover search algo- rithm to find the optimum value of an objective func- tion. Thisisachievedbyfl...

  2. [2]

    cost" Hamiltonian (equa- tion 15) and Hm is the

    QAOA-SCF QAOA, or the Quantum Approximate Optimization Algorithm [3], has emerged as a prominent strategy for solving combinatorial optimization problems on near- term quantum hardware. In the context of this work, a QAOA-SCF algorithm can be written as: EQAOA-SCF = min R min ⃗ g,⃗b D ⃗ g,⃗b Hc(R) ⃗ g,⃗b E (16) where the QAOA anstaz is defined as: ⃗ g,⃗b ...

  3. [3]

    The inner optimization of Equation (15) defines an Ising problem

    QA-SCF QuantumAnnealing(QA)isaninnovativeformofana- log computation designed to utilize quantum mechanical effects to search for the optimal solutions of Ising prob- lems. The inner optimization of Equation (15) defines an Ising problem. Therefore, the QA-SCF algorithm simply solves the inner optimization via Quantum Annealing, followed by updating the mo...

  4. [4]

    It ex- ploits the sparse Fourier spectrum of specific objective functions, reducing the optimization task to a decoding problem

    DQI-SCF Decoded Quantum Interferometry (DQI) is a quantum algorithm designed to solve combinatorial optimization problems by finding approximate solutions [6]. It ex- ploits the sparse Fourier spectrum of specific objective functions, reducing the optimization task to a decoding problem. The MaxCut SCF problem of Equation (E12) can be converted to a weigh...

  5. [5]

    Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. Sawaya,et al., Chemical reviews 119, 10856 (2019)

    2019

  6. [6]

    Abbas, A

    A. Abbas, A. Ambainis, B. Augustino, A. Bärtschi, H. Buhrman, C. Coffrin, G. Cortiana, V. Dunjko, D. J. Egger, B. G. Elmegreen,et al., Nature Reviews Physics , 1 (2024)

    2024

  7. [7]

    A quan- tum approximate optimization algorithm,

    E. Farhi, J. Goldstone, and S. Gutmann, “A quan- tum approximate optimization algorithm,” (2014), arXiv:1411.4028 [quant-ph]

    arXiv 2014

  8. [8]

    Apolloni, C

    B. Apolloni, C. Carvalho, and D. De Falco, Stochastic Processes and their Applications33, 233 (1989)

    1989

Show all 122 references
  1. [9]

    Apolloni, N

    B. Apolloni, N. Cesa-Bianchi, and D. De Falco, in Stochastic Processes, Physics and Geometry: Proceed- ings of the Ascona-Locarno Conference(1990) pp. 97– 111

    1990

  2. [10]

    S. P. Jordan, N. Shutty, M. Wootters, A. Zalcman, A. Schmidhuber, R. King, S. V. Isakov, and R. Bab- bush, arXiv preprint arXiv:2408.08292 (2024)

    2024

  3. [11]

    L. K. Grover, inProceedings of the Twenty-Eighth An- nual ACM Symposium on Theory of Computing, STOC ’96 (Association for Computing Machinery, New York, NY, USA, 1996) p. 212–219

    1996

  4. [12]

    B. W. Reichardt, inProceedings of the Thirty-Sixth An- nual ACM Symposium on Theory of Computing, STOC ’04 (Association for Computing Machinery, New York, NY, USA, 2004) p. 502–510

    2004

  5. [13]

    Boros, Y

    E. Boros, Y. Crama, and E. Rodríguez-Heck, Journal of combinatorial optimization39, 687 (2020)

    2020

  6. [14]

    T. M. Alkhamis, M. Hasan, and M. A. Ahmed, Euro- pean journal of operational research108, 641 (1998)

    1998

  7. [15]

    Barahona, M

    F. Barahona, M. Jünger, and G. Reinelt, Mathematical programming 44, 127 (1989)

    1989

  8. [16]

    P. M. Pardalos and G. P. Rodgers, Computing45, 131 (1990)

    1990

  9. [17]

    Helmberg and F

    C. Helmberg and F. Rendl, Mathematical programming 82, 291 (1998)

    1998

  10. [18]

    H. Goto, K. Tatsumura, and A. R. Dixon, Science ad- vances 5, eaav2372 (2019)

    2019

  11. [19]

    Kadowaki and H

    T. Kadowaki and H. Nishimori, Physical Review E58, 5355 (1998)

    1998

  12. [20]

    Matsubara, M

    S. Matsubara, M. Takatsu, T. Miyazawa, T. Shibasaki, Y. Watanabe, K. Takemoto, and H. Tamura, in2020 25th Asia and South Pacific Design Automation Con- ference (ASP-DAC)(IEEE, 2020) pp. 667–672

    2020

  13. [21]

    Yamaoka, C

    M. Yamaoka, C. Yoshimura, M. Hayashi, T. Okuyama, H. Aoki, and H. Mizuno, in 2015 IEEE Interna- tional Solid-State Circuits Conference-(ISSCC) Digest of Technical Papers(IEEE, 2015) pp. 1–3

    2015

  14. [22]

    Honjo, T

    T. Honjo, T. Sonobe, K. Inaba, T. Inagaki, T. Ikuta, Y. Yamada, T. Kazama, K. Enbutsu, T. Umeki, R. Kasahara, et al., Science advances 7, eabh0952 (2021)

    2021

  15. [23]

    Aspuru-Guzik, A

    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science309, 1704 (2005)

    2005

  16. [24]

    D. R. Hartree, inMathematical Proceedings of the Cam- bridge Philosophical Society,Vol.24(Cambridgeuniver- sity press, 1928) pp. 89–110

    1928

  17. [25]

    Fock, Zeitschrift für Physik61, 126 (1930)

    V. Fock, Zeitschrift für Physik61, 126 (1930)

    1930

  18. [26]

    C. C. J. Roothaan, Reviews of modern physics23, 69 (1951)

    1951

  19. [27]

    G. G. Hall, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences205, 541 (1951)

    1951

  20. [28]

    J. A. Pople and D. L. Beveridge,Approximate Molecu- lar Orbital Theory, Advanced Chemistry (McGraw Hill Higher Education, Maidenhead, England, 1970)

    1970

  21. [29]

    Szabo and N

    A. Szabo and N. S. Ostlund,Modern Quantum Chem- istry: Introduction to Advanced Electronic Structure Theory, Dover Books on Chemistry (Dover Publica- tions, Mineola, NY, 1996)

    1996

  22. [30]

    Helgaker, P

    T. Helgaker, P. Jorgensen, and J. Olsen, Molecular electronic-structure theory(John Wiley & Sons, 2013)

    2013

  23. [31]

    Lehtola, International Journal of Quantum Chem- istry 119, e25968 (2019)

    S. Lehtola, International Journal of Quantum Chem- istry 119, e25968 (2019)

    2019

  24. [32]

    Cross, Applied Mathematical Modelling 9, 226 (1985)

    M. Cross, Applied Mathematical Modelling 9, 226 (1985)

    1985

  25. [33]

    Cancès and C

    E. Cancès and C. Le Bris, ESAIM: Mathematical Mod- elling and Numerical Analysis34, 749 (2000)

    2000

  26. [34]

    J. D. Whitfield and Z. Zimborás, The Journal of chemical physics 141 (2014), https://doi.org/10.1063/1.4903453

    2014 doi

  27. [35]

    J. D. Whitfield, N. Schuch, and F. Verstraete, Many- Electron Approaches in Physics, Chemistry and Math- ematics: A Multidisciplinary View , 245 (2014)

    2014

  28. [36]

    Farhi, S

    E. Farhi, S. Gutmann, D. Ranard, and B. Villalonga, arXiv preprint arXiv:2503.12789 (2025)

    2025

  29. [37]

    R. Xia, T. Bian, and S. Kais, The Journal of Physical Chemistry B 122, 3384 (2018)

    2018

  30. [38]

    Copenhaver, A

    J. Copenhaver, A. Wasserman, and B. Wehefritz- Kaufmann, The Journal of Chemical Physics 154 (2021), https://doi.org/10.1063/5.0030397

    2021 doi

  31. [39]

    D. A. Chermoshentsev, A. O. Malyshev, M. Esen- can, E. S. Tiunov, D. Mendoza, A. Aspuru-Guzik, A. K. Fedorov, and A. I. Lvovsky, arXiv preprint arXiv:2106.13167 (2021)

    2021 arXiv

  32. [40]

    S. N. Genin, I. G. Ryabinkin, and A. F. Izmaylov, arXiv preprint arXiv:1901.04715 (2019)

    2019 arXiv

  33. [41]

    Okada, M

    S. Okada, M. Ohzeki, M. Terabe, and S. Taguchi, Sci- entific reports 9, 2098 (2019)

    2019

  34. [42]

    Linderberg and Y

    J. Linderberg and Y. Öhrn, International Journal of Quantum Chemistry 12, 161 (1977)

    1977

  35. [43]

    Dalgaard and P

    E. Dalgaard and P. Jo/rgensen, The Journal of Chemi- cal Physics 69, 3833 (1978)

    1978

  36. [44]

    Couty and M

    M. Couty and M. B. Hall, The Journal of Physical Chemistry A 101, 6936 (1997)

    1997

  37. [45]

    Kossoski, A

    F. Kossoski, A. Marie, A. Scemama, M. Caffarel, and P.-F. Loos, Journal of Chemical Theory and Computa- tion 17, 4756 (2021)

    2021

  38. [46]

    K. R. Glaesemann and M. W. Schmidt, The Journal of Physical Chemistry A114, 8772 (2010)

    2010

  39. [47]

    A. T. Wong and R. J. Harrison, Journal of computa- tional chemistry 16, 1291 (1995)

    1995

  40. [48]

    D. J. Thouless, Nuclear Physics21, 225 (1960)

    1960

  41. [49]

    Čížek and J

    J. Čížek and J. Paldus, The Journal of Chemical Physics 47, 3976 (1967)

    1967

  42. [50]

    Paldus and J

    J. Paldus and J. Čĺžzek, The Journal of Chemical Physics 52, 2919 (1970)

    1970

  43. [51]

    Seeger and J

    R. Seeger and J. A. Pople, The Journal of Chemical Physics 66, 3045 (1977). 15

    1977

  44. [52]

    Bauernschmitt and R

    R. Bauernschmitt and R. Ahlrichs, The Journal of chemical physics 104, 9047 (1996)

    1996

  45. [53]

    Dehareng and G

    D. Dehareng and G. Dive, Journal of Computational Chemistry 21, 483 (2000)

    2000

  46. [54]

    S. M. Sharada, D. Stück, E. J. Sundstrom, A. T. Bell, and M. Head-Gordon, Molecular Physics 113, 1802 (2015)

    2015

  47. [55]

    Kohn and L

    W. Kohn and L. J. Sham, Physical review140, A1133 (1965)

    1965

  48. [56]

    M. X. Goemans and D. P. Williamson, inProceedings of the twenty-sixth annual ACM symposium on Theory of computing(1994) pp. 422–431

    1994

  49. [57]

    Khot, in Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC ’02 (Association for Computing Machinery, New York, NY, USA, 2002) p

    S. Khot, in Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC ’02 (Association for Computing Machinery, New York, NY, USA, 2002) p. 767–775

    2002

  50. [58]

    S. Khot, G. Kindler, E. Mossel, and R. O’Donnell, SIAM Journal on Computing37, 319 (2007)

    2007

  51. [59]

    Wolpert and W

    D. Wolpert and W. Macready, IEEE Transactions on Evolutionary Computation 1, 67 (1997)

    1997

  52. [60]

    Van Lenthe, R

    J. Van Lenthe, R. Zwaans, H. J. Van Dam, and M. Guest, Journal of computational chemistry27, 926 (2006)

    2006

  53. [61]

    A. C. Vaucher and M. Reiher, Journal of chemical the- ory and computation13, 1219 (2017)

    2017

  54. [62]

    Lehtola, Journal of chemical theory and computation 15, 1593 (2019)

    S. Lehtola, Journal of chemical theory and computation 15, 1593 (2019)

    2019

  55. [63]

    Nesterov, Optimization methods and software9, 141 (1998)

    Y. Nesterov, Optimization methods and software9, 141 (1998)

    1998

  56. [64]

    Charikar and A

    M. Charikar and A. Wirth, in45th Annual IEEE Sym- posium on Foundations of Computer Science(2004) pp. 54–60

    2004

  57. [65]

    M. X. Goemans and D. P. Williamson, Journal of the ACM (JACM) 42, 1115 (1995)

    1995

  58. [66]

    Alon and A

    N. Alon and A. Naor, inProceedings of the thirty-sixth annual ACM symposium on Theory of computing(2004) pp. 72–80

    2004

  59. [67]

    B. O. Roos, P. R. Taylor, and P. E. Sigbahn, Chemical Physics 48, 157 (1980)

    1980

  60. [68]

    G. A. Quantum, Collaborators*†, F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley,et al., Science 369, 1084 (2020)

    2020

  61. [69]

    Gilliam, S

    A. Gilliam, S. Woerner, and C. Gonciulea, Quantum5, 428 (2021)

    2021

  62. [70]

    Bulger, W

    D. Bulger, W. P. Baritompa, and G. R. Wood, Journal ofoptimizationtheoryandapplications 116,517(2003)

    2003

  63. [71]

    W. P. Baritompa, D. W. Bulger, and G. R. Wood, SIAM Journal on Optimization15, 1170 (2005)

    2005

  64. [72]

    Blekos, D

    K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, Physics Reports1068, 1 (2024)

    2024

  65. [73]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.- Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, Nature communications5, 4213 (2014)

    2014

  66. [74]

    Q. Sun, X. Zhang, S. Banerjee, P. Bao, M. Barbry, N. S. Blunt, N. A. Bogdanov, G. H. Booth, J. Chen, Z.-H. Cui, et al., The Journal of chemical physics153 (2020), https://doi.org/10.1063/5.0006074

    2020 doi

  67. [75]

    W. J. Hehre, R. Ditchfield, and J. A. Pople, The Jour- nal of Chemical Physics56, 2257 (1972)

    1972

  68. [76]

    T. H. Dunning Jr, The Journal of chemical physics90, 1007 (1989)

    1989

  69. [77]

    Wigner and P

    E. Wigner and P. Jordan, Z. Phys47, 46 (1928)

    1928

  70. [78]

    dwave-tabu,

    “dwave-tabu,” https://github.com/dwavesystems/ dwave-tabu (2024)

    2024

  71. [79]

    dwave-neal,

    “dwave-neal,” https://github.com/dwavesystems/ dwave-neal (2024)

    2024

  72. [80]

    qiskit-optimization,

    “qiskit-optimization,” https://github.com/ qiskit-community/qiskit-optimization (2024)

    2024

  73. [81]

    Quantum computing with Qiskit,

    A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, “Quantum computing with Qiskit,” (2024), arXiv:2405.08810 [quant-ph]

    2024 arXiv

  74. [82]

    Diamond and S

    S. Diamond and S. Boyd, Journal of Machine Learning Research 17, 1 (2016)

    2016

  75. [83]

    C. R. Harris, K. J. Millman, S. J. Van Der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith,et al., Nature 585, 357 (2020)

    2020

  76. [84]

    Nocedal and S

    J. Nocedal and S. J. Wright, Numerical optimization , 101 (2006)

    2006

  77. [85]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Peter- son, W. Weckesser, J. Bright,et al., Nature methods 17, 261 (2020)

    2020

  78. [86]

    D. G. Smith, L. A. Burns, A. C. Simmon- ett, R. M. Parrish, M. C. Schieber, R. Galvelis, P. Kraus, H. Kruse, R. Di Remigio, A. Alenaizan, et al., The Journal of chemical physics 152 (2020), https://doi.org/10.1063/5.0006002

    2020 doi

  79. [87]

    M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scal- mani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Or- tiz, A. F. Izmay...

    2016

  80. [88]

    12,e1606(2022)

    F.Neese,WIRESComput.Molec.Sci. 12,e1606(2022)

    2022

  81. [89]

    Palmieri, R

    P. Palmieri, R. Tarroni, and S. Rettrup, The Journal of chemical physics100, 5849 (1994)

    1994

  82. [90]

    S. B. Bravyi and A. Y. Kitaev, Annals of Physics298, 210 (2002)

    2002

  83. [91]

    J. T. Seeley, M. J. Richard, and P. J. Love, The Journal of chemical physics 137 (2012), https://doi.org/10.1063/1.4768229

    2012 doi

  84. [92]

    J.R.McClean, J.Romero, R.Babbush, andA.Aspuru- Guzik, New Journal of Physics18, 023023 (2016)

    2016

  85. [93]

    Chen and H

    W. Chen and H. B. Schlegel, The Journal of chemical physics 101, 5957 (1994)

    1994

  86. [94]

    Yamada and S

    T. Yamada and S. Hirata, The Journal of Chemical Physics 143(2015), https://doi.org/10.1063/1.4929354. 16

    2015 doi

  87. [95]

    T.-C. Yen, R. A. Lang, and A. F. Izmaylov, The Journal of chemical physics 151 (2019), https://doi.org/10.1063/1.5110682

    2019 doi

  88. [96]

    J. S. Andrews, D. Jayatilaka, R. G. Bone, N. C. Handy, and R. D. Amos, Chemical physics letters 183, 423 (1991)

    1991

  89. [97]

    Kempe, A

    J. Kempe, A. Kitaev, and O. Regev, Siam journal on computing 35, 1070 (2006)

    2006

  90. [98]

    S.P.JordanandE.Farhi,PhysicalReviewA 77,062329 (2008)

    2008

  91. [99]

    Bravyi, D

    S. Bravyi, D. P. DiVincenzo, D. Loss, and B. M. Terhal, Physical review letters101, 070503 (2008)

    2008

  92. [100]

    Babbush, B

    R. Babbush, B. O’Gorman, and A. Aspuru-Guzik, An- nalen der Physik525, 877 (2013)

    2013

  93. [101]

    J.Bausch,in Annales Henri Poincaré,Vol.21(Springer,

  94. [102]

    Cichy, P

    S. Cichy, P. K. Faehrmann, S. Khatri, and J. Eisert, Phys. Rev. A109, 052624 (2024)

    2024

  95. [103]

    R. I. G, Centre d’Études de Recherche Opérationnelle 17, 71 (1975)

    1975

  96. [104]

    J. D. Biamonte, Physical Review A77, 052331 (2008), arXiv:0801.3800

    2008 arXiv

  97. [105]

    Boros and P

    E. Boros and P. L. Hammer, Discrete Applied Mathe- matics 123, 155 (2002)

    2002

  98. [106]

    Buchheim and G

    C. Buchheim and G. Rinaldi, SIAM Journal on Opti- mization 18, 1398 (2008)

    2008

  99. [107]

    Elloumi, A

    S. Elloumi, A. Lambert, and A. Lazare, Journal of Global Optimization 80, 231 (2021), arXiv:1901.07904

    2021 arXiv

  100. [108]

    Tanburn, E

    R. Tanburn, E. Okada, and N. S. Dattani, arXiv preprint arXiv:1411.6758 (2014)

    2014 arXiv

  101. [109]

    Boros and P

    E. Boros and P. L. Hammer, Annals of Operations Re- search 33, 151 (1991)

    1991

  102. [110]

    Babbush, J

    R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Physical Review A91, 022311 (2015)

    2015

  103. [111]

    Snir and R

    S. Snir and R. Yuster, SIAM Journal on Computing41, 1466 (2012)

    2012

  104. [112]

    Chatziafratis, M

    V. Chatziafratis, M. Mahdian, and S. Ahmadian, in Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Proceedings of Ma- chineLearningResearch, Vol.130,editedbyA.Banerjee and K. Fukumizu (PMLR, 2021) pp. 1657–1665

    2021

  105. [113]

    P. R. Taylor, International journal of quantum chem- istry 31, 521 (1987)

    1987

  106. [114]

    Pulay, Chemical physics letters73, 393 (1980)

    P. Pulay, Chemical physics letters73, 393 (1980). 17 APPENDIX Appendix A: Change of Basis In this appendix, we derive how Equation (3) is obtained from similarity transformed ladder operators. Noteˆk is defined in Equation (2) in the main text. First, we write the following: [...

    1980

  107. [115]

    X K∈χ Y i∈K I − Zi 2 − Ω #2 = λ

    Penalty T erms We rewrite the QUSO optimization problem with penalty terms as: EVP QU SO= min R min |b⟩∈D ⟨b| HD(R) + λVP |b⟩ . (B2) Here VP penalizes any Slater determinant not in the setS by λ, which takes a large positive value. This provides a way to favour certain molecul...

    1975

  108. [116]

    Select an arbitrary cubic termxixjxk

  109. [117]

    Notexa is an extra ancillary bit

    Replace withxixjxk 7→ xaxk + xixj − 2xixa − xjxa + 3xa, yielding a(k − 1)-local function. Notexa is an extra ancillary bit

  110. [118]

    Alternate transformations have been also been given such as by Boros and Hammer in [101, Section 4.4], Buchheim and Rinaldi in [102] and Elloumi et al

    Repeat steps 1 and 2 (in totalk − 2 times) until the resulting function is2-local. Alternate transformations have been also been given such as by Boros and Hammer in [101, Section 4.4], Buchheim and Rinaldi in [102] and Elloumi et al. in [103]. We note that one advantage of qu...

  111. [119]

    As this problem is optimized over single Fock states, we replace eachZi operator with a spin variablezi ∈ {−1, +1}

    QUSO to QUBO mapping In the main text it is shown that that diagonal Hamiltonian (equation 4) is mapped to a QUSO under the Jordan Wigner transform: 24 HD(R) = MX n ¯hnn I − Zn 2 + 1 2 MX m,n m̸=n ¯gmmnn − ¯gmnnm 4 I − Zn − Zm + ZmZn 7→ HQU SO(R) = MX n ¯hnn 1 − zn 2 + 1 2 MX ...

  112. [120]

    We use this procedure to inspire a mapping for the QUSO problem

    QUSO to MaxCut mapping In [105, Section 3] Boros and Hammer show how a QUBO problem can be mapped onto a Max-Cut problem using a single ancilla/auxilary bit. We use this procedure to inspire a mapping for the QUSO problem. The MaxCut cost function is defined as follows: CM axC...

  113. [121]

    Table VII and Table VIII provide the numerical results obtained from the SCF algorithms developed in this work

    OH – data Table V provides the numerical SCF results for the OH– anion performed in conventional chemistry software. Table VII and Table VIII provide the numerical results obtained from the SCF algorithms developed in this work. Psi4 PySCF Guassian Guassian (QC) ORCA PySCF (SO...

  114. [122]

    Tables XII,XIIIand XIV provide the CISD numerical results

    N 2 data Tables IX, X and XI provide the SCF results for the potential energy surface of molecular Nitgrogen. Tables XII,XIIIand XIV provide the CISD numerical results. Figure 4 graphically shows these CISD results. The RHF and SO-SCF results are obtained from a calculation pe...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.