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Symmetry Dilemmas in Quantum Computing for Chemistry: A Comprehensive Analysis

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Pith's one-line read The gate-efficient pool saGSpD (singlet spin-adapted singles plus perfect-pairing doubles) is proven non-universal once point-group symmetry is enforced; simulations map when symmetry-breaking pools collapse to the wrong state.

desk verdict The ≥4-irrep non-universality theorem is solid and new, but the advertised two-irrep generalization is asserted rather than proven; the paper deserves refereeing and a requested revision either completing or scoping that claim. read the letter →

arxiv 2512.12097 v1 pith:7APAJNH5 submitted 2025-12-13 quant-ph physics.chem-ph

Ilias Magoulas , Muhan Zhang , Francesco A. Evangelista This is my paper · ORCID
classification quant-phphysics.chem-ph
keywords symmetry-adaptedoperatorpoolsADAPT-VQEuniversalitypoint-groupsymmetryspin-adaptedsinglesperfect-pairingdoublesLiealgebravariationalcollapse
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

Quantum simulations of molecules build variational wavefunctions from operator pools, and the most circuit-efficient spin-preserving pool in wide use — singlet spin-adapted singles plus perfect-pairing doubles (saGSpD) — stops being universal the moment molecular point-group symmetry is enforced. The proof runs through a parity invariant: every allowed excitation preserves the even-or-odd count of electrons within each symmetry class of orbitals, so families of double excitations present in the universal parent pool are unreachable, and with them certain exact-state components. The claim is proven outright for point groups with four or more symmetry classes, argued via explicit nested commutators for two-class groups, and confirmed numerically: the adaptive variational eigensolver with the symmetry-enforced pool never attains chemical accuracy for linear H6, and exactly the two predicted symmetry-adapted basis states remain orthogonal to the converged wavefunction. Companion simulations map a practical rule — with the target state the global ground state, universal symmetry-breaking pools eventually self-correct; when crossing states differ in several symmetries, conserving any one of them suffices; when they differ in a single symmetry, only conserving that one prevents variational collapse. The result matters because it makes a genuine three-way trade-off explicit for the most economical pool family: strict symmetry adaptation, universal expressiveness, and shallow circuits cannot be had simultaneously, and benchmarks that report only energies can miss the deficiency.

What carries the argument

The central object is a parity invariant (Eq. 16): for each orbital symmetry class (irreducible representation), a sign recording whether that class holds an even or odd number of electrons. Spin-adapted singles move one electron within a class and perfect-pairing doubles move two, so every generator — hence every commutator and unitary — preserves every parity. This blocks the double excitations the universal parent pool relies on: four-distinct-class operators are unreachable (a complete proof for groups with four or more classes), and the triplet-intermediate excitation with matched class pairs arises only inside the locked combination √3A[1] − A[0] (two-class groups, by explicit nested c

What would settle it

Build the Lie algebra numerically for a small two-class system (say a four-orbital model with Cs symmetry): write the symmetry-enforced saGSpD generators as explicit matrices, close the algebra under commutation until the dimension stabilizes, and test whether any triplet-intermediate double excitation A[1]^{RS}_{PQ} with Γ_P = Γ_Q ≠ Γ_R = Γ_S lies in the span — if yes, the two-class claim is false. Or run ADAPT-VQE-saGSpD on H6 with a larger basis such as 6-31G: the theorem predicts the analogous basis states stay orthogonal and chemical accuracy is never reached, while populating those state

Watch

Extended reading notes

Core claim

The paper's central claim: the saGSpD pool — singlet spin-adapted singles plus perfect-pairing doubles — is not universal once restricted to totally symmetric operators of a non-trivial point group. Every generator preserves the electron-count parity of each orbital symmetry class, hence the whole Lie algebra does, putting many totally symmetric double excitations out of reach: proven for groups with at least four classes, argued via nested commutators for two-class groups. ADAPT-VQE with the enforced pool stagnates on linear H6 without chemical accuracy; the two predicted basis states stay orthogonal throughout. Dropping one symmetry restores universality, but sacrificing the wrong one coll

Load-bearing premise

For point groups with only two symmetry classes (Cs, Ci, C2), the theorem that the missing triplet-intermediate excitations can never be generated rests on the assertion that the commutator families enumerated up to third nesting in the supplementary material, together with a structural argument about number-operator strings, exhaust the Lie algebra — stated without an inductive closure proof or a dimension count, so the two-class claim is only as strong as that enumeration.

Editorial extensions

If this is right

  • Any variational ansatz built from totally symmetric singlet singles plus perfect-pairing doubles cannot reach the exact symmetry-adapted wavefunction for a molecule with non-trivial point-group symmetry: ADAPT-VQE with the enforced pool never attains chemical accuracy for linear H6, and exactly two of the 92 symmetry-adapted basis states remain orthogonal at every geometry.
  • Universal pools that drop a single symmetry converge to numerically exact energies in the tested cases (GSD and saGSpD-full on H6 and CH2), with symmetry contamination appearing early and self-correcting — but only when the target state is the global minimum or another distinguishing symmetry is retained.
  • The safe-use rule: when two crossing states differ in several symmetry properties, conserving any one of them suffices to avoid collapse (both GSD and saGSpD-full succeed for CH2's A1–B1 crossing); when they differ in one property only, that property must be conserved — GSD collapses to the triplet in BeH2 and saGSpD-full collapses to the Π² state in BO.
  • Fully symmetry-adapted universal pools are the most parameter-efficient (pDint0 reaches chemical accuracy in 67 operators for H6) but cost the most per operator — hundreds to thousands of CNOT gates — making the three-way trade-off between universality, symmetry preservation, and gate count explicit.
  • The paper claims the phenomenon is generic: any quantum algorithm that uses symmetry-violating gates risks symmetry contamination and variational collapse, so the three-scenario rule applies beyond the specific pools simulated here.

Reading between the lines

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

  • Extension: the parity-invariant argument is a transferable diagnostic — any operator pool whose generators change each symmetry class's electron count by an even amount is confined to a fixed parity sector, so other truncated pools (seniority-truncated or number-operator-weighted excitations) can be screened for expressivity ceilings up front using the same invariant.
  • The paper leaves open whether the two-class outcome is a theorem: a Lie-algebra dimension calculation for a minimal two-class Fock space would either promote the Cs/Ci/C2 claim to a proof by showing the enumeration closes, or expose a higher-order commutator that isolated the missing excitation.
  • A sharper test follows from the paper's own data: at stretched H6 geometries the large error shrinks because several states become degenerate, so energy-error benchmarks systematically underestimate the deficiency; a molecule whose correlation is dominated by the missing excitations should make the pool fail far more catastrophically than the 24–36 mEh seen here.
  • The paper's fortuitously exact BeH2 case implies that minimum-basis benchmarks of non-universal pools are optimistic by construction — enlarging the basis to add orbitals of the missing symmetry classes should convert that accidental success into a failure, a prediction the authors do not test.
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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

2 major / 5 minor

Summary. The paper analyzes trade-offs among symmetry adaptation, universality, and gate efficiency in operator pools for variational quantum eigensolvers, focusing on the singlet spin-adapted singles plus perfect-pairing doubles (saGSpD) pool. The central theoretical claim is that enforcing spatial symmetry in saGSpD, for molecules with non-trivial point-group symmetry, makes the pool non-universal: certain singlet spin-adapted double excitations cannot be generated by the Lie algebra of the totally symmetric saGSpD pool. The paper proves this rigorously for elementary Abelian 2-groups with at least four irreps using a parity invariant Π_I (Eq. 16), and argues the same for two-irrep groups (Ci, C2, Cs) using commutator enumeration. The numerical part performs ADAPT-VQE simulations with four pools (GSD, saGSpD-full, symmetry-adapted saGSpD, pDint0) in three physically motivated scenarios: a global ground state (H6), crossing states differing in multiple symmetries (CH2), and crossing states differing in one symmetry (BeH2 for spin, BO for spatial symmetry). The resulting guidelines concern when symmetry-breaking pools are safe, when one distinguishing symmetry suffices, and when a specific symmetry must be enforced.

Significance. If the full theorem is correct, the paper identifies a fundamental and practically relevant limitation of a widely used gate-efficient ansatz pool, and the numerical scenarios provide a useful benchmark map for symmetry-related variational collapse. The ≥4-irrep part of the proof is clean and convincing: the parity invariant is preserved by every saGSpD generator and hence by the whole Lie algebra, so parity-flipping double excitations are provably unreachable. The paper also gives concrete, reproducible numerical evidence (CSF overlaps, symmetry weights, potential energy curves) across four molecular systems. The main weakness is that the two-irrep case, which is needed for the full force of the abstract and for the H6 stagnation demonstration, is supported by an enumerated commutator analysis rather than an exhaustive closure argument. The numerical simulations are extensive and the paper is clearly written, but the central theorem as stated is not fully proved.

major comments (2)
  1. [Section III, Eqs. (17)-(18) and Supplement S1] The proof of non-universality for two-irrep point groups (Ci, C2, Cs) is incomplete. The argument shows one family of nested commutators that produces A[1]RS_PQ only with the irrep restrictions Γ_P=Γ_S and Γ_Q=Γ_R (row-1 pattern), and then asserts that A[1]RS_PQ with Γ_P=Γ_Q≠Γ_R=Γ_S (row-2 pattern) is absent. Supplement S1 explicitly states that the enumeration goes only 'up to triply nested commutators' and provides no inductive closure proof, dimension count, or invariant separating row-2 A[1] from the generated algebra. In fact, Eq. (S22) shows [[A^QQ_PP, A^R_Q], A^S_P] = A[0]^QR_PS, which by relabeling gives an in-algebra A[0]^RS_PQ with exactly the row-2 irrep pattern. Thus the algebra contains the A[0] needed to disentangle a √3A[1]−A[0] combination, and whether some higher-order commutator yields the corresponding A[1] or the combination for row 2 is not settled. This gap is load-
  2. [Section III and Table II] The paper asserts that saGSpD-full—the saGSpD pool without spatial-symmetry enforcement—is universal ('when non-totally symmetric operators are incorporated into it, the saGSpD pool is universal'), and Table II lists it as universal. No proof or reference for this assertion is provided. This matters because the numerical taxonomy and the safety guidelines for 'symmetry-breaking but universal pools' rely on saGSpD-full being universal, and its failure in the BO case (Sec. V.C) is interpreted as symmetry breaking rather than as possible non-universality. Please either supply a proof of universality (e.g., showing the Lie algebra generated by all spin-adapted singles and perfect-pairing doubles closes on the full symmetry-adapted saGSD algebra) or cite a reference that establishes it.
minor comments (5)
  1. [Abstract and Section III] The abstract states that the saGSpD pool is non-universal for 'point groups other than C1', but the proof is restricted to elementary Abelian 2-groups (Ci, C2, Cs, D2, C2v, C2h, D2h). The argument for perfect-pairing doubles being totally symmetric by nature relies on the 2-group property Γ⊗Γ = 1. The abstract and conclusions should be qualified to this class, or the proof should be extended to cover cyclic and non-Abelian groups that appear in molecular point-group practice.
  2. [Eq. (18)] The proportionality in Eq. (18) is stated without the constant; it would be clearer to display the exact factor (1/√2 or 1/2) and specify the assumed occupancies (P doubly occupied, S unoccupied) explicitly in the equation or immediately before it.
  3. [Table I] The condition 'ξ = Q if R = P, otherwise ξ = R' in the header of Table I is cryptic. Please define ξ in the caption or footnote, and consider giving an example for each of the five rows.
  4. [Section IV] The saGSpD-full pool is defined only as 'the spatial-symmetry-violating variant'; the specific extension of ADAPT-VQE with tuples of two and three non-totally symmetric singles should be described more explicitly, including why these tuples are sufficient for universality and how the closed-form expression of Ref. 48 is used.
  5. [Section V.A] The sentence 'At that limit, all four states are degenerate and any linear combination of them is an eigenstate' is correct, but the preceding discussion of the 3.0 Å H6 result would benefit from a quantitative tie-in to the orthogonal CSF weights shown in Fig. S9, since the text references but does not interpret the figure in the main body.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the non-universality theorem rests on a self-contained parity invariant and commutator analysis, and the numerical checks are benchmarked against FCI rather than fitted.

full rationale

The central derivation in Section III starts from the explicitly defined pool generators (Eqs. 13–15), constructs the parity invariant Π_I (Eq. 16), and shows by direct parity and nested-commutator reasoning that certain totally symmetric double-excitation operators cannot belong to the Lie algebra generated by the spatially symmetrized saGSpD elements. This is a mathematical argument from stated definitions, not an output fitted to an input. The H6/STO-6G simulations (Figs. S4–S9) test the theorem against FCI overlaps: exactly the two predicted CSFs with irrep pattern Γ_0=Γ_2, Γ_3=Γ_5, Γ_0≠Γ_3 remain orthogonal, so this is an externally falsifiable check rather than a renamed conclusion. Self-citations (e.g., ref. 9 for pDint0 and circuit costs, ref. 59 for QForte) are contextual or are independently validated by FCI benchmarks in this paper; they are not the sole warrant for the non-universality claim. The only caveat is the two-irrep (Ci, C2, Cs) case, where Supplement S1 enumerates commutators up to triple nesting rather than giving an inductive closure proof; that is a proof-completeness/rigor concern, not circularity, because the missing piece would be additional mathematical argument, not a restatement of the conclusion as an input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to produce the central claims; the theorem is parameter-free and the numerics are benchmarked against FCI/STO-6G as an external standard. The listed 'free parameters' are optimization hyperparameters that steer which operators ADAPT-VQE selects but do not enter the universality theorem. The axis-loading is the two-irrep Lie-algebra exhaustiveness claim (axiom 4) and the self-cited universality of saGSD/pDint0 (axiom 2), the latter being independent support from a companion paper rather than circular input. No new physical entities are postulated; the saGSpD-full tuple construction is a methodological variant rather than an invented entity.

free parameters (3)
  • SHGO sampling points (Sobol, 32) = 32
    Hand-chosen number of sampling points for the simplicial-homology global optimization used only in the saGSpD-full operator-selection step (Section IV); affects which operators are selected but not the qualitative conclusions.
  • Basin-hopping settings for BO/saGSpD-full = 30 iterations, temperature 1e-4 Eh
    Global-optimization settings needed because naive BFGS did not converge for BO with the saGSpD-full pool (Section IV); they alter optimization trajectories, not the pools' structural properties.
  • ADAPT-VQE convergence thresholds = 10^-6 Eh gradient norm; 2000 micro-iterations; stop at FCI dimension minus 1
    Convergence and termination criteria chosen by hand (Section IV); generous enough that saGSpD stagnation is unlikely to be an optimizer artifact, but the paper does not scan thresholds to demonstrate this.
assumptions (5)
  • standard math Standard molecular point groups (Ci, C2, Cs, D2, C2v, C2h, D2h) are elementary Abelian 2-groups, so irrep labels combine like Z2 vectors
    Invoked at the start of Section III to justify the parity arguments and the Table I conditions; standard group theory.
  • domain assumption The GSD pool with repeated elements is universal (ref 38) and the saGSD/pDint0 pools are universal (ref 9)
    The 'target' Lie algebra the proof compares against is the universal symmetry-adapted pool algebra. Ref 9 is by the same authors (Magoulas & Evangelista 2025), a companion derivation not machine-checked; however, the negative non-universality claim does not depend on it, since the parity invariant shows absence regardless.
  • domain assumption Reference states are restricted HF (closed- or open-shell) Slater determinants defining the symmetry sectors
    Used throughout Section IV; the reachable-space analysis (e.g., the two orthogonal H6 CSFs) is specific to the RHF reference |Φ⟩ and the chosen orbital ordering.
  • ad hoc to paper For two-irrep point groups, spin-polarized double excitations can only be generated multiplied by number-operator strings, and the displayed commutator families exhaust the relevant part of the Lie algebra
    Section III argument for Ci/C2/Cs; asserted with enumeration up to triply nested commutators (Supp. S1, Eqs. S40–S49) rather than a complete inductive or dimension-counting proof. This is the weakest premise (see weakest_assumption).
  • standard math Thouless theorem: a singles-only operator set can only rotate within the reference mean-field manifold
    Cited (ref 47) in Section III to explain the known fully-spin-polarized limitation of saGSpD.

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Pith. "Pith review of Symmetry Dilemmas in Quantum Computing for Chemistry: A Comprehensive Analysis." pith.science (2026). https://pith.science/paper/7APAJNH5

@misc{pith2026251212097,
  author       = {Pith},
  title        = {Pith review of: Symmetry Dilemmas in Quantum Computing for Chemistry: A Comprehensive Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7APAJNH5}},
  note         = {Machine review of arXiv:2512.12097}
}
read the original abstract

Symmetry adaptation, universality, and gate efficiency are central but often competing requirements in quantum algorithms for electronic structure and many-body physics. For example, fully symmetry-adapted universal operator pools typically generate long and deep quantum circuits, gate-efficient universal operator pools generally break symmetries, and gate-efficient fully symmetry-adapted operator pools may not be universal. In this work, we analyze such symmetry dilemmas both theoretically and numerically. On the theory side, we prove that the popular, gate-efficient operator pool consisting of singlet spin-adapted singles and perfect-pairing doubles is not universal when spatial symmetry is enforced. To demonstrate the strengths and weaknesses of the three types of pools, we perform numerical simulations using an adaptive algorithm paired with operator pools that are (i) fully symmetry-adapted and universal, (ii) fully symmetry-adapted and non-universal, and (iii) breaking a single symmetry and are universal. Our numerical simulations encompass three physically relevant scenarios in which the target state is (i) the global ground state, (ii) the ground state crossed by a state differing in multiple symmetry properties, and (iii) the ground state crossed by a state differing in a single symmetry property. Our results show when symmetry-breaking but universal pools can be used safely, when enforcing at least one distinguishing symmetry suffices, and when a particular symmetry must be rigorously preserved to avoid variational collapse. Together, the formal and numerical analysis provides a practical guide for designing and benchmarking symmetry-adapted operator pools that balance universality, resource requirements, and robust state targeting in quantum simulations for chemistry.

Figures

Figures reproduced from arXiv: 2512.12097 by the authors.

Figure 1
Figure 1. FIG. 1. Quantum circuit performing a Givens rotation by an angle [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The two lowest-energy potential energy curves for the systems of interest, computed at the FCI/STO-6G level of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Convergence of ADAPT-VQE/STO-6G simulations [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Convergence of ADAPT-VQE/STO-6G simulations using the GSD, saGSpD-full, saGSpD, and pDint0 operator pools [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Convergence of ADAPT-VQE/STO-6G simulations using the GSD, saGSpD-full, saGSpD, and pDint0 operator pools [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of ADAPT-VQE/STO-6G simulations using the GSD, saGSpD-full, saGSpD, and pDint0 operator pools [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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Works this paper leans on

63 extracted references · 14 canonical work pages

  1. [1]

    author author D. E. \ Deutsch ,\ title title Quantum computational networks , \ https://doi.org/10.1098/rspa.1989.0099 journal journal Proc. R. Soc. Lond. A \ volume 425 ,\ pages 73--90 ( year 1989 ) NoStop

    arXiv 1989

  2. [2]

    author author D. E. \ Deutsch , author A. Barenco ,\ and\ author A. Ekert ,\ title title Universality in quantum computation , \ https://doi.org/10.1098/rspa.1995.0065 journal journal Proc. R. Soc. Lond. A \ volume 449 ,\ pages 669--677 ( year 1995 ) NoStop

    arXiv 1995

  3. [3]

    author author D. P. \ DiVincenzo ,\ title title Two-bit gates are universal for quantum computation , \ https://doi.org/10.1103/PhysRevA.51.1015 journal journal Phys. Rev. A \ volume 51 ,\ pages 1015--1022 ( year 1995 ) NoStop

    doi:10.1103/physreva.51.1015 1995

  4. [4]

    Barenco , author C

    author author A. Barenco , author C. H. \ Bennett , author R. Cleve , author D. P. \ DiVincenzo , author N. Margolus , author P. Shor , author T. Sleator , author J. A. \ Smolin ,\ and\ author H. Weinfurter ,\ title title Elementary gates for quantum computation , \ https://doi.org/10.1103/PhysRevA.52.3457 journal journal Phys. Rev. A \ volume 52 ,\ pages...

    doi:10.1103/physreva.52.3457 1995

  5. [5]

    author author P. O. \ Boykin , author T. Mor , author M. Pulver , author V. Roychowdhury ,\ and\ author F. Vatan ,\ title title On universal and fault-tolerant quantum computing: A novel basis and a new constructive proof of universality for S hor's basis , \ in\ https://doi.org/10.1109/SFFCS.1999.814621 booktitle Proc. 40th Annu. Symp. Found. Comput. Sci...

    arXiv 1999

  6. [6]

    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 ,\ title title A variational eigenvalue solver on a photonic quantum processor , \ https://doi.org/10.1038/ncomms5213 journal journal Nat. Commun. \ volume 5 ,\ pages 42...

    doi:10.1038/ncomms5213 2014

  7. [7]

    author author A. Y. \ Kitaev ,\ @noop title Quantum measurements and the A belian stabilizer problem , \ note 1995, arXiv:quant-ph/9511026. arXiv.org e-Print archive. https://arxiv.org/abs/quant-ph/9511026 NoStop

    arXiv 1995

  8. [8]

    author author Y. S. \ Yordanov , author D. R. M. \ Arvidsson-Shukur ,\ and\ author C. H. W. \ Barnes ,\ title title Efficient quantum circuits for quantum computational chemistry , \ https://doi.org/10.1103/PhysRevA.102.062612 journal journal Phys. Rev. A \ volume 102 ,\ pages 062612 ( year 2020 ) NoStop

    doi:10.1103/physreva.102.062612 2020

Show all 63 references
  1. [9]

    Magoulas \ and\ author F

    author author I. Magoulas \ and\ author F. A. \ Evangelista ,\ @noop title Spin-adapted fermionic unitaries: F rom L ie algebras to compact quantum circuits , \ note 2025, arXiv:2511.13485. arXiv.org e-Print archive. https://arxiv.org/abs/2511.13485 NoStop

    2025 arXiv

  2. [10]

    author author I. G. \ Ryabinkin , author T.-C. \ Yen , author S. N. \ Genin ,\ and\ author A. F. \ Izmaylov ,\ title title Qubit coupled cluster method: A systematic approach to quantum chemistry on a quantum computer , \ https://doi.org/10.1021/acs.jctc.8b00932 journal journa...

    2018 doi

  3. [11]

    author author I. G. \ Ryabinkin , author R. A. \ Lang , author S. N. \ Genin ,\ and\ author A. F. \ Izmaylov ,\ title title Iterative qubit coupled cluster approach with efficient screening of generators , \ https://doi.org/10.1021/acs.jctc.9b01084 journal journal J. Chem. The...

    2020 doi

  4. [12]

    author author H. L. \ Tang , author V. O. \ Shkolnikov , author G. S. \ Barron , author H. R. \ Grimsley , author N. J. \ Mayhall , author E. Barnes ,\ and\ author S. E. \ Economou ,\ title title Qubit-ADAPT-VQE : An adaptive algorithm for constructing hardware-efficient ans\"...

    2021 doi

  5. [13]

    Sun , author X

    author author Z. Sun , author X. Li , author J. Liu , author Z. Li ,\ and\ author J. Yang ,\ title title Circuit-efficient qubit excitation-based variational quantum eigensolver , \ https://doi.org/10.1021/acs.jctc.5c00119 journal journal J. Chem. Theory Comput. \ volume 21 ,\...

    2025 doi

  6. [14]

    Ram\^ o a , author P

    author author M. Ram\^ o a , author P. G. \ Anastasiou , author L. P. \ Santos , author N. J. \ Mayhall , author E. Barnes ,\ and\ author S. E. \ Economou ,\ title title Reducing the resources required by ADAPT-VQE using coupled exchange operators and improved subroutines , \ ...

    2025 doi

  7. [15]

    author author M. Nooijen ,\ title title Can the eigenstates of a many-body Hamiltonian be represented exactly using a general two-body cluster expansion? \ https://doi.org/10.1103/PhysRevLett.84.2108 journal journal Phys. Rev. Lett. \ volume 84 ,\ pages 2108--2111 ( year 2000 ) NoStop

    doi

  8. [16]

    Nakatsuji ,\ title title Structure of the exact wave function , \ https://doi.org/10.1063/1.1287275 journal journal J

    author author H. Nakatsuji ,\ title title Structure of the exact wave function , \ https://doi.org/10.1063/1.1287275 journal journal J. Chem. Phys. \ volume 113 ,\ pages 2949--2956 ( year 2000 ) NoStop

    doi

  9. [17]

    author author G.-L. R. \ Anselmetti , author D. Wierichs , author C. Gogolin ,\ and\ author R. M. \ Parrish ,\ title title Local, expressive, quantum-number-preserving VQE ans\" a tze for fermionic systems , \ https://doi.org/10.1088/1367-2630/ac2cb3 journal journal New J. Phy...

    2021 doi

  10. [18]

    author author H. G. A. \ Burton , author D. Marti-Dafcik , author D. P. \ Tew ,\ and\ author D. J. \ Wales ,\ title title Exact electronic states with shallow quantum circuits from global optimisation , \ https://doi.org/10.1038/s41534-023-00744-2 journal journal npj Quantum I...

    2023 doi

  11. [19]

    author author H. G. A. \ Burton ,\ title title Accurate and gate-efficient quantum ans\" a tze for electronic states without adaptive optimization , \ https://doi.org/10.1103/PhysRevResearch.6.023300 journal journal Phys. Rev. Research \ volume 6 ,\ pages 023300 ( year 2024 ) NoStop

    2024 doi

  12. [20]

    Lykos \ and\ author G

    author author P. Lykos \ and\ author G. W. \ Pratt ,\ title title Discussion on the H artree- F ock approximation , \ https://doi.org/10.1103/revmodphys.35.496 journal journal Rev. Mod. Phys. \ volume 35 ,\ pages 496--501 ( year 1963 ) NoStop

    1963 doi

  13. [21]

    author author H. R. \ Grimsley , author S. E. \ Economou , author E. Barnes ,\ and\ author N. J. \ Mayhall ,\ title title An adaptive variational algorithm for exact molecular simulations on a quantum computer , \ https://doi.org/10.1038/s41467-019-10988-2 journal journal Nat ...

    2019 doi

  14. [22]

    author author Y. S. \ Yordanov , author V. Armaos , author C. H. W. \ Barnes ,\ and\ author D. R. M. \ Arvidsson-Shukur ,\ title title Qubit-excitation-based adaptive variational quantum eigensolver , \ https://doi.org/10.1038/s42005-021-00730-0 journal journal Commun. Phys. \...

    2021 doi

  15. [23]

    Coester ,\ title title Bound states of a many-particle system , \ https://doi.org/10.1016/0029-5582(58)90280-3 journal journal Nucl

    author author F. Coester ,\ title title Bound states of a many-particle system , \ https://doi.org/10.1016/0029-5582(58)90280-3 journal journal Nucl. Phys. \ volume 7 ,\ pages 421--424 ( year 1958 ) NoStop

    1958 doi

  16. [24]

    Coester \ and\ author H

    author author F. Coester \ and\ author H. K \" u mmel ,\ title title Short-range correlations in nuclear wave functions , \ https://doi.org/10.1016/0029-5582(60)90140-1 journal journal Nucl. Phys. \ volume 17 ,\ pages 477--485 ( year 1960 ) NoStop

    1960 doi

  17. [25]

    C \' z ek ,\ title title On the correlation problem in atomic and molecular systems

    author author J. C \' z ek ,\ title title On the correlation problem in atomic and molecular systems. Calculation of wavefunction components in Ursell -type expansion using quantum-field theoretical methods , \ https://doi.org/10.1063/1.1727484 journal journal J. Chem. Phys. \...

    1966 doi

  18. [26]

    author author J. C \' z ek ,\ title title On the use of the cluster expansion and the technique of diagrams in calculations of correlation effects in atoms and molecules , \ https://doi.org/10.1002/9780470143599.ch2 journal journal Adv. Chem. Phys. \ volume 14 ,\ pages 35--89 ...

    1969 doi

  19. [27]

    Kutzelnigg ,\ in\ https://doi.org/10.1007/978-1-4757-0887-5_5 booktitle Methods of Electronic Structure Theory ,\ editor edited by\ editor H

    author author W. Kutzelnigg ,\ in\ https://doi.org/10.1007/978-1-4757-0887-5_5 booktitle Methods of Electronic Structure Theory ,\ editor edited by\ editor H. F. \ Schaefer , III \ ( publisher Springer ,\ address Boston ,\ year 1977 )\ pp.\ pages 129--188 NoStop

    1977 doi

  20. [28]

    Kutzelnigg ,\ title title Quantum chemistry in Fock space

    author author W. Kutzelnigg ,\ title title Quantum chemistry in Fock space. I. The universal wave and energy operators , \ https://doi.org/10.1063/1.444231 journal journal J. Chem. Phys. \ volume 77 ,\ pages 3081--3097 ( year 1982 ) NoStop

    1982 doi

  21. [29]

    author author R. J. \ Bartlett , author S. A. \ Kucharski ,\ and\ author J. Noga ,\ title title Alternative coupled-cluster ans\" a tze II. The unitary coupled-cluster method , \ https://doi.org/10.1016/S0009-2614(89)87372-5 journal journal Chem. Phys. Lett. \ volume 155 ,\ pa...

    1989 doi

  22. [30]

    author author P. G. \ Szalay , author M. Nooijen ,\ and\ author R. J. \ Bartlett ,\ title title Alternative ans\" a tze in single reference coupled-cluster theory. III. A critical analysis of different methods , \ https://doi.org/10.1063/1.469641 journal journal J. Chem. Phys....

    1995 doi

  23. [31]

    author author A. G. \ Taube \ and\ author R. J. \ Bartlett ,\ title title New perspectives on unitary coupled-cluster theory , \ https://doi.org/10.1002/qua.21198 journal journal Int. J. Quantum Chem. \ volume 106 ,\ pages 3393--3401 ( year 2006 ) NoStop

    2006 doi

  24. [32]

    Cooper \ and\ author P

    author author B. Cooper \ and\ author P. J. \ Knowles ,\ title title Benchmark studies of variational, unitary and extended coupled cluster methods , \ https://doi.org/10.1063/1.3520564 journal journal J. Chem. Phys. \ volume 133 ,\ pages 234102 ( year 2010 ) NoStop

    2010 doi

  25. [33]

    author author F. A. \ Evangelista ,\ title title Alternative single-reference coupled cluster approaches for multireference problems: The simpler, the better , \ https://doi.org/10.1063/1.3598471 journal journal J. Chem. Phys. \ volume 134 ,\ pages 224102 ( year 2011 ) NoStop

    2011 doi

  26. [34]

    Harsha , author T

    author author G. Harsha , author T. Shiozaki ,\ and\ author G. E. \ Scuseria ,\ title title On the difference between variational and unitary coupled clutser theories , \ https://doi.org/10.1063/1.5011033 journal journal J. Chem. Phys. \ volume 148 ,\ pages 044107 ( year 2018 ) NoStop

    2018 doi

  27. [35]

    \ Filip \ and\ author A

    author author M.-A. \ Filip \ and\ author A. J. W. \ Thom ,\ title title A stochastic approach to unitary coupled cluster , \ https://doi.org/10.1063/5.0026141 journal journal J. Chem. Phys. \ volume 153 ,\ pages 214106 ( year 2020 ) NoStop

    2020 doi

  28. [36]

    author author J. K. \ Freericks ,\ title title Operator relationship between conventional coupled cluster and unitary coupled cluster , \ https://doi.org/10.3390/sym14030494 journal journal Symmetry \ volume 14 ,\ pages 494 ( year 2022 ) NoStop

    2022 doi

  29. [37]

    Anand , author P

    author author A. Anand , author P. Schleich , author S. Alperin-Lea , author P. W. K. \ Jensen , author S. Sim , author M. D\' i az-Tinoco , author J. S. \ Kottmann , author M. Degroote , author A. F. \ Izmaylov ,\ and\ author A. Aspuru-Guzik ,\ title title A quantum computing...

    2022 doi

  30. [38]

    author author F. A. \ Evangelista , author G. K.-L. \ Chan ,\ and\ author G. E. \ Scuseria ,\ title title Exact parameterization of fermionic wave functions via unitary coupled cluster theory , \ https://doi.org/10.1063/1.5133059 journal journal J. Chem. Phys. \ volume 151 ,\ ...

    2019 doi

  31. [39]

    Paldus , author B

    author author J. Paldus , author B. G. \ Adams ,\ and\ author J. C \' i z ek ,\ title title Application of graphical methods of spin algebras to limited CI approaches. I . closed shell case , \ https://doi.org/10.1002/qua.560110511 journal journal Int. J. Quantum Chem. \ volum...

    1977 doi

  32. [40]

    Paldus ,\ title title Correlation problems in atomic and molecular systems

    author author J. Paldus ,\ title title Correlation problems in atomic and molecular systems. V . spin-adapted coupled cluster many-electron theory , \ https://doi.org/10.1063/1.434526 journal journal J. Chem. Phys. \ volume 67 ,\ pages 303 ( year 1977 ) NoStop

    1977 doi

  33. [41]

    author author B. G. \ Adams \ and\ author J. Paldus ,\ title title Orthogonally-spin-adapted coupled-cluster theory for closed-shell systems including triexcited clusters , \ https://doi.org/10.1103/PhysRevA.20.1 journal journal Phys. Rev. A \ volume 20 ,\ pages 1 ( year 1979 ) NoStop

    1979 doi

  34. [42]

    author author R. A. \ Chiles \ and\ author C. E. \ Dykstra ,\ title title An electron pair operator approach to coupled cluster wave functions. application to H e _2 , B e _2 , and M g _2 and comparison with CEPA methods , \ https://doi.org/10.1063/1.441643 journal journal J. ...

    1981 doi

  35. [43]

    Piecuch \ and\ author J

    author author P. Piecuch \ and\ author J. Paldus ,\ title title Orthogonally spin-adapted coupled-cluster equations involving singly and doubly excited clusters. C omparison of different procedures for spin-adaptation , \ https://doi.org/10.1002/qua.560360402 journal journal I...

    1989 doi

  36. [44]

    Geertsen , author S

    author author J. Geertsen , author S. Eriksen ,\ and\ author J. Oddershede ,\ title title Some aspects of the coupled cluster based polarization propagator method , \ https://doi.org/10.1016/S0065-3276(08)60364-0 journal journal Adv. Quantum Chem. \ volume 22 ,\ pages 167--209...

    1991 doi

  37. [45]

    Piecuch \ and\ author J

    author author P. Piecuch \ and\ author J. Paldus ,\ title title Orthogonally spin-adapted multi-reference H ilbert space coupled-cluster formalism: diagrammatic formulation , \ https://doi.org/10.1007/BF01113244 journal journal Theor. Chim. Acta \ volume 83 ,\ pages 69 ( year ...

    1992 doi

  38. [46]

    Piecuch \ and\ author J

    author author P. Piecuch \ and\ author J. Paldus ,\ title title Orthogonally spin-adapted state-universal coupled-cluster formalism: Implementation of the complete two-reference theory including cubic and quartic coupling terms , \ https://doi.org/10.1063/1.467304 journal jour...

    1994 doi

  39. [47]

    author author D. J. \ Thouless ,\ title title Stability conditions and nuclear rotations in the H artree-- F ock theory , \ https://doi.org/10.1016/0029-5582(60)90048-1 journal journal Nucl. Phys. \ volume 21 ,\ pages 225 ( year 1960 ) NoStop

    1960 doi

  40. [48]

    Magoulas \ and\ author F

    author author I. Magoulas \ and\ author F. A. \ Evangelista ,\ title title Closed-form expressions for unitaries of spin-adapted operators , \ https://doi.org/10.1080/00268976.2025.2534672 journal journal Mol. Phys. ,\ pages e2534672 ( year published online 2025 ) NoStop

    2025

  41. [49]

    Endres , author C

    author author S. Endres , author C. Sandrock ,\ and\ author W. Focke ,\ title title A simplicial homology algorithm for L ipschitz optimisation , \ https://doi.org/10.1007/s10898-018-0645-y journal journal J. Glob. Optim. \ volume 72 ( year 2018 ),\ 10.1007/s10898-018-0645-y NoStop

    2018 doi

  42. [50]

    Virtanen , author R

    author author P. Virtanen , author R. Gommers , author T. E. \ Oliphant , author M. Haberland , author T. R. \ amd D. Cournapeau , author E. Burovski , author P. Peterson , author W. Weckesser , author J. Bright , author S. J. \ van der Walt , author M. Brett , author J. Wilso...

    2020

  43. [51]

    Joe \ and\ author F

    author author S. Joe \ and\ author F. Y. \ Kuo ,\ title title Constructing S obol sequences with better two-dimensional projections , \ https://doi.org/10.1137/070709359 journal journal SIAM J. Sci. Comput. \ volume 30 ,\ pages 2635--2654 ( year 2008 ) NoStop

    2008 doi

  44. [52]

    author author C. G. \ Broyden ,\ title title The convergence of a class of double-rank minimization algorithms: 2. The new algorithm , \ https://doi.org/10.1093/imamat/6.3.222 journal journal IMA J. Appl. Math. \ volume 6 ,\ pages 222--231 ( year 1970 ) NoStop

    1970 doi

  45. [53]

    Fletcher ,\ title title A new approach to variable metric algorithms , \ https://doi.org/10.1093/comjnl/13.3.317 journal journal Comput

    author author R. Fletcher ,\ title title A new approach to variable metric algorithms , \ https://doi.org/10.1093/comjnl/13.3.317 journal journal Comput. J. \ volume 13 ,\ pages 317--322 ( year 1970 ) NoStop

    1970 doi

  46. [54]

    Goldfarb ,\ title title A family of variable-metric methods derived by variational means , \ https://doi.org/10.1090/S0025-5718-1970-0258249-6 journal journal Math

    author author D. Goldfarb ,\ title title A family of variable-metric methods derived by variational means , \ https://doi.org/10.1090/S0025-5718-1970-0258249-6 journal journal Math. Comp. \ volume 24 ,\ pages 23--26 ( year 1970 ) NoStop

    1970 doi

  47. [55]

    author author D. F. \ Shanno ,\ title title Conditioning of quasi- Newton methods for function minimization , \ https://doi.org/10.1090/S0025-5718-1970-0274029-X journal journal Math. Comp. \ volume 24 ,\ pages 647--656 ( year 1970 ) NoStop

    1970 doi

  48. [56]

    Wales ,\ title title Energy landscapes and properties of biomolecules , \ https://doi.org/10.1088/1478-3975/2/4/S02 journal journal Phys

    author author D. Wales ,\ title title Energy landscapes and properties of biomolecules , \ https://doi.org/10.1088/1478-3975/2/4/S02 journal journal Phys. Biol. \ volume 2 ,\ pages S86--93 ( year 2005 ) NoStop

    2005 doi

  49. [57]

    author author D. G. A. \ Smith , author L. A. \ Burns , author A. C. \ Simmonett , author R. M. \ Parrish , author M. C. \ Schieber , author R. Galvelis , author P. Kraus , author H. Kruse , author R. D. \ Remigio , author A. Alenaizan , author A. M. \ James , author S. Lehtol...

    2020

  50. [58]

    author author W. J. \ Hehre , author R. F. \ Stewart ,\ and\ author J. A. \ Pople ,\ title title Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater -type atomic orbitals , \ https://doi.org/10.1063/1.1672392 journal journal J. Chem. Phys. \ volu...

    1969 doi

  51. [59]

    author author N. H. \ Stair \ and\ author F. A. \ Evangelista ,\ title title QForte : An efficient state-vector emulator and quantum algorithms library for molecular electronic structure , \ https://doi.org/10.1021/acs.jctc.1c01155 journal journal J. Chem. Theory Comput. \ vol...

    2022 doi

  52. [60]

    author author T. M. \ Henderson , author I. W. \ Bulik , author T. Stein ,\ and\ author G. E. \ Scuseria ,\ title title Seniority-based coupled cluster theory , \ https://doi.org/10.1063/1.4904384 journal journal J. Chem. Phys. \ volume 141 ,\ pages 244104 ( year 2014 ) NoStop

    2014 doi

  53. [61]

    Magoulas \ and\ author F

    author author I. Magoulas \ and\ author F. A. \ Evangelista ,\ title title Unitary coupled cluster: S eizing the quantum moment , \ https://doi.org/10.1021/acs.jpca.3c02781 journal journal J. Phys. Chem. A \ volume 127 ,\ pages 6567--6576 ( year 2023 ) NoStop

    2023 doi

  54. [62]

    author author A. F. \ Izmaylov ,\ title title On construction of projection operators , \ https://doi.org/10.1021/acs.jpca.9b01103 journal journal J. Phys. Chem. A \ volume 123 ,\ pages 3429--3433 ( year 2019 ) NoStop

    2019 doi

  55. [63]

    Kuroiwa \ and\ author Y

    author author K. Kuroiwa \ and\ author Y. O. \ Nakagawa ,\ title title Penalty methods for a variational quantum eigensolver , \ https://doi.org/10.1103/PhysRevResearch.3.013197 journal journal Phys. Rev. Research \ volume 3 ,\ pages 013197 ( year 2021 ) NoStop

    2021 doi

Pith tools

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