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Grassmann Variational Monte Carlo with neural wave functions

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper establishes that excited states of the two-dimensional Heisenberg model can be computed to about $10^{-4}$ relative error by performing variational Monte Carlo over a whole linear subspace of Hilbert space.

desk verdict A worthwhile geometric formalization of excited-state VMC with clean 6x6 benchmarks; the unexamined conditioning of per-sample overlap matrices is the main soft spot, but not a fatal one. read the letter →

arxiv 2507.10287 v1 pith:ITVQKOQW submitted 2025-07-14 quant-ph

Douglas Hendry , Alessandro Sinibaldi , Giuseppe Carleo This is my paper · ORCID
classification quant-ph
keywords excitedstatesGrassmannianneuralquantumvariationalMonteCarlostochasticreconfigurationHeisenbergmodelmany-body
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 accurate computation of excited states no longer requires symmetry constraints or penalty terms if the variational object is promoted from a single wavefunction to an $N$-dimensional subspace. It formalizes the determinant-sampling excited-state framework of [1] in the language of Grassmann geometry, turning the search for several excited states into one simultaneous geometric optimization. It generalizes Stochastic Reconfiguration to multiple wavefunctions and introduces matrix versions of variances, overlaps, and fidelities, all estimated by a single determinant sampling scheme. On the 2D Heisenberg model the method yields relative energy errors around $10^{-4}$ on a $6\times6$ lattice and V-scores of order $10^{-3}$ on $10\times10$, with spin structure factors matching benchmarks.

What carries the argument

The central object is the Grassmannian $\mathrm{Gr}_N(H)$, the manifold of $N$-dimensional linear subspaces of the Hilbert space, realized by the span of $N$ neural wavefunctions $\Phi=(\phi_1,\dots,\phi_N)$. The carrying identity is the wedge-product overlap $\det[[\Psi|\Phi]]=\langle\langle\Psi|\Phi\rangle\rangle$, which turns a subspace into a single antisymmetric wavefunction and motivates determinant sampling, $P_\Phi(S)\propto |\det[[S|\Phi]]|^2/(N!\det G)$. The local operator matrix $\tilde A(S)=\Phi^{-1}(S)A(S)$ is the estimator that avoids constructing and inverting the Gram matrix directly. The geometric optimization uses the Grassmann Quantum Geometric Tensor $S_{\mu\nu}=\frac{1}{N}\mathrm{Tr}\big(G^{-1}[[\partial_\mu\Phi|\hat P_{V^\perp}|\partial_\nu\Phi]]\big)$, whose inversion defines the Stochastic Reconfiguration update; operator covariance matrices built the same way supply variances and observables such as the spin structure factor.

What would settle it

Run Grassmann VMC on an $8\times8$ Heisenberg lattice in a degenerate momentum sector while recording the smallest singular value of each sampled overlap matrix $\Phi(S)$; if a noticeable fraction of samples are singular or ill-conditioned, the energy and variance estimators would become unstable or biased, which would show the claimed estimator is not well-defined as stated.

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Extended reading notes

Core claim

Central claim: Grassmann Variational Monte Carlo yields accurate excited states of many-body systems by optimizing a linear subspace of the Hilbert space, not a single wavefunction. The subspace $V$ is represented by a basis of $N$ neural wavefunctions $\Phi=(\phi_1,\dots,\phi_N)$, and the wedge-product identity $\det[[\Psi|\Phi]]=\langle\langle\Psi|\Phi\rangle\rangle$ lets the whole subspace be sampled as a single antisymmetric state over $N$-tuples of configurations with probability $P_\Phi(S)\propto |\det[[S|\Phi]]|^2/(N!\det G)$. All quantum values are promoted from scalars to matrices: the operator expectation matrix is $\tilde A(\Phi)=G^{-1}[[\Phi|A|\Phi]]$, its eigenvalues are the approximate excited-state energies, and its Monte Carlo estimator is the average of local matrices $\tilde A(S)=\Phi^{-1}(S)A(S)$. The same covariance estimators yield operator variance matrices, overlap and fidelity matrices between subspaces, and the Grassmann Quantum Geometric Tensor that defines Stochastic Reconfiguration. On the $6\times6$ Heisenberg model the first four excited states in tested momentum and spin-flip sectors reach relative energy errors at or below about $10^{-4}$; on $10\times10$ the V-scores are of order $10^{-3}$ and the ground-state energy density $-0.671544(4)$ agrees with quantum Monte Carlo.

Load-bearing premise

The load-bearing premise is that every sampled N-tuple of configurations gives an invertible overlap matrix, because the local operator estimator divides by that matrix, and the sampling distribution does not exclude near-singular tuples.

Editorial extensions

If this is right

  • On a $6\times6$ Heisenberg lattice, the first four excited states in every tested momentum and spin-flip sector have relative energy errors at or below about $10^{-4}$ against exact diagonalization.
  • On a $10\times10$ lattice, six excited states per sector are obtained with V-scores of order $10^{-3}$ or better, and the ground-state energy and $S(\pi,\pi)$ agree with quantum Monte Carlo.
  • Excited-state spin structure factors are read off directly from the diagonal of the operator variance matrix, so a single optimization run supplies both energies and correlation observables.
  • Stochastic Reconfiguration extends naturally to simultaneous optimization of several wavefunctions, so multiple eigenstates are produced in one run without penalty terms or hand-imposed symmetry constraints.
  • The formulation is written in second quantization and purely in terms of the Hamiltonian, which opens the same treatment to lattice fermions and frustrated spin systems.

Reading between the lines

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

  • I infer that the determinant-sampling estimator will need a practical singularity monitor in production use: one should track the smallest singular value of the sampled overlap matrix $\Phi(S)$, and near-degenerate sectors may require regularization or a pseudo-inverse, since the paper does not address this case.
  • I expect the same subspace geometry to transfer to ab initio electronic structure, where the wedge-product picture matches Slater determinants; a natural test is whether simultaneous excited-state optimization with neural backflow surpasses single-reference quantum chemistry on small molecules.
  • I see a natural route to real-time dynamics through the new subspace overlap and fidelity matrices: rather than evolving a single state, one could project time evolution onto a moving Grassmann subspace and monitor leakage through the operator variance, a direction the authors mention through subspace-expansion methods.
  • I read the main computational bottleneck as the $N^2$ forward passes needed for the overlap matrix entries when using separate networks; shared-backbone architectures mitigate this, and further architectural sharing would determine how large $N$ can practically become.
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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

3 major / 5 minor

Summary. The paper formalizes the excited-state variational framework of Pfau et al. in terms of Grassmann geometry of Hilbert space. It promotes scalar quantum values (expectation values, variances, covariances, fidelities) to N×N matrices with basis-independent eigenvalues, defines a Grassmann quantum geometric tensor, and extends stochastic reconfiguration to the simultaneous optimization of N variational wave functions. The method is tested on the 2D Heisenberg model: on a 6×6 lattice the first four excited states in many momentum/spin-flip sectors are compared with exact diagonalization and found to have relative energy errors around or below 10^-4, and on a 10×10 lattice six excited states are reported with V-scores of order 10^-3 or below. Spin structure factors are also computed for both system sizes, with the ground-state values checked against a quantum Monte Carlo reference.

Significance. If the claims hold, this is a valuable and timely contribution: it gives a clean geometric language for multi-state variational Monte Carlo, generalizes stochastic reconfiguration and operator-variance estimators to subspaces, and provides a practical route to excited states of two-dimensional spin systems where tensor-network or sign-problem-based methods struggle. The paper's strengths are its careful derivations of determinant-sampling averages in the appendices, the clean 6×6 exact-diagonalization benchmark across many symmetry sectors, and the explicit comparison of the 10×10 ground-state energy and structure factor with quantum Monte Carlo. The main gap is that the 10×10 excited states are validated only through internal V-scores rather than independent benchmarks, and the determinant-sampling estimator has no conditioning analysis.

major comments (3)
  1. [Sec. III.B.3, Eq. (13)] The local operator matrix eA(S)=Φ^{-1}(S)·A(S) requires the sampled N×N overlap matrix Φ(S) to be invertible, and the sampling probability PΦ(S)∝|det Φ(S)|^2/(N! det G) only excludes exactly singular tuples. A tuple with det Φ(S)~ε contributes probability O(ε²) but local matrix elements O(1/ε), so the estimator can have very large fluctuations or undefined values if near-singular tuples occur with non-negligible frequency. The paper neither proves finite variance over the finite configuration space nor reports condition-number or determinant diagnostics from the runs. Because the OVM/OCM estimates in Eq. (14), the V-scores in Fig. 3/Table II, and the spin structure factors in Sec. V all rely on these local matrices, this is load-bearing for the central numerical claim. Please add a conditioning analysis, a regularized inverse, or empirical determinant/condition-number diagnostics, or prove that the estimator has finite variance for the models considered.
  2. [Sec. V, Fig. 3 and Table II] The 10×10 excited-state results are supported only by V-scores, which are an internal consistency measure. The statement that for a given V-score the actual energy relative error is typically an order of magnitude smaller is a heuristic from Ref. [42] and is not established for excited states of this model; the ground state is checked against QMC, but no excited state has an independent benchmark. To support the abstract claim of 'highly accurate energies and physical observables for a large number of excited states', please provide external validation for at least a subset of the 10×10 excited states (e.g., DMRG or a finite-size-scaling comparison), or explicitly temper the accuracy claim to what V-scores can certify.
  3. [Sec. I vs. Sec. III.B.3] The claim that the method 'avoids to invert the possibly ill-conditioned Gram matrix of the basis states' is overstated. Equation (13) still requires inversion of the N×N overlap matrix Φ(S) for every sampled tuple, and near-singularity of Φ(S) is exactly the kind of ill-conditioning that the text says is bypassed. Please clarify the precise sense in which the method avoids Gram-matrix inversion and discuss the relationship between the conditioning of G and the conditioning of Φ(S).
minor comments (5)
  1. [Sec. VII] The software statement says the code 'will be made public in a later revision'; for a numerical-methods paper, full reproducibility would be helped by releasing the code or at least providing per-run hyperparameters, sample counts, and random seeds in the final version.
  2. [Sec. III.B.5] The sentence 'the ground state of H with fermionic symmetry is the Slater determinant of the first N excited states' is ambiguous; it should read 'the N lowest-energy eigenstates' (including the ground state) to avoid implying that the ground state is built from the N-th through 2N-th levels.
  3. [Fig. 2 and Fig. 3] The figures mix two spin-flip sectors (qsf=0,1) but the marker legend is implicit; please add explicit labels or a legend so the reader can identify which marker corresponds to which sector.
  4. [Sec. II.B] There is a typo in 'the the bra matrix of its dual basis'.
  5. [Appendix 2, Eq. (34)] The summation notation over S^{N-M} is introduced only in passing; please state explicitly whether the sums run over ordered tuples or sets and how the (N-M)! prefactor in Eq. (34) arises from that convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central numerical claims are benchmarked against independent exact diagonalization and quantum Monte Carlo results, and the Grassmann framework is explicitly presented as a formalization of prior work by Pfau et al., not as an independent derivation of those results.

full rationale

The paper's core numerical claim is the accurate computation of excited states of the 2D Heisenberg model, validated externally on 6x6 lattices against exact diagonalization (Table I and Fig. 2) and on 10x10 lattices against independent quantum Monte Carlo ground-state energy and spin structure factor values (Figs. 3 and 4). These benchmarks are not fitted inputs: the variational energies are reported with Monte Carlo uncertainties and compared to independent data, so the claim does not reduce to a fitted parameter renamed as a prediction. The Grassmann formalism is explicitly an extension of the framework of Pfau et al. (Ref. [1]), as stated in the abstract: 'we rigorously formalize the framework introduced by Pfau et al. in terms of Grassmann geometry of the Hilbert space.' This is an acknowledged relationship to prior work, not a concealed import. The new estimator formulas, including the operator covariance matrices in Eq. (14), are derived in Appendices 2 and 3 from standard linear-algebra minor identities, and the sampling estimator Eq. (13) is derived as an exact Monte Carlo identity rather than assumed to match the target energies. Self-citations to NetKet and to prior neural quantum state literature are implementation and background support, not load-bearing steps in the validation. The conditioning concern about near-singular overlap matrices Phi(S) in Eq. (13) is a legitimate numerical robustness question, but it is a correctness risk, not a circularity: the estimator is not defined in terms of the energies it claims to predict. No step in the derivation chain reduces by construction to its own output, and no fitted input is relabeled as a prediction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities; it is a mathematical and algorithmic reformulation. The main inputs are standard linear algebra, the variational ansatz architecture, and the assumption of linear independence. The hyperparameters are chosen by hand but are not fitted to the benchmark energies.

free parameters (4)
  • Learning rate = 0.15/Ns
    Set by hand for SR updates; affects convergence speed and stability but not the target result.
  • QGT diagonal shift = 10^-3
    Added to stabilize the Quantum Geometric Tensor inversion in SR; chosen without systematic tuning.
  • Monte Carlo samples and SR steps = 2^11 samples, about 5000 steps
    Chosen to balance statistical accuracy and computational cost; higher values would improve precision.
  • Network hyperparameters = C=4/6, d=(4,6,12)/(8,12,20), B=(2,2,8), RBM hidden 16/12, K=3, Kdw=5, f=4
    Architecture sizes chosen by hand following prior neural quantum state works; they determine representational capacity.
assumptions (4)
  • domain assumption The N variational wave functions are linearly independent so that they span an N-dimensional subspace (Sec. IV.A).
    The variational construction assumes the VWFs form a basis; if they become linearly dependent, the Gram matrix is singular and the formalism breaks.
  • standard math The determinant sampling scheme is a valid Monte Carlo estimator with the given probabilities and averages (Eqs. 11-14, Appendix 2-3).
    The derivations use matrix minor identities; these are standard but the provided appendix text is incomplete in places.
  • domain assumption Marshall sign rule is applicable to the Heisenberg model and is applied to the neural network outputs (Sec. V).
    The sign rule reduces the sign problem for the bipartite Heisenberg model; it is an input to the ansatz.
  • domain assumption The CNN+RBM ansatz with imposed translational and spin-flip symmetries is expressive enough to represent the target excited states in each sector (Sec. V).
    The accuracy of the results depends on the ansatz being close to the true eigenstates in the relevant symmetry sectors; this is validated a posteriori by energies and V-scores.

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Cite this review

Pith. "Pith review of Grassmann Variational Monte Carlo with neural wave functions." pith.science (2026). https://pith.science/paper/ITVQKOQW

@misc{pith2026250710287,
  author       = {Pith},
  title        = {Pith review of: Grassmann Variational Monte Carlo with neural wave functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ITVQKOQW}},
  note         = {Machine review of arXiv:2507.10287}
}
read the original abstract

Excited states play a central role in determining the physical properties of quantum matter, yet their accurate computation in many-body systems remains a formidable challenge for numerical methods. While neural quantum states have delivered outstanding results for ground-state problems, extending their applicability to excited states has faced limitations, including instability in dense spectra and reliance on symmetry constraints or penalty-based formulations. In this work, we rigorously formalize the framework introduced by Pfau et al.~\cite{pfau2024accurate} in terms of Grassmann geometry of the Hilbert space. This allows us to generalize the Stochastic Reconfiguration method for the simultaneous optimization of multiple variational wave functions, and to introduce the multidimensional versions of operator variances and overlaps. We validate our approach on the Heisenberg quantum spin model on the square lattice, achieving highly accurate energies and physical observables for a large number of excited states.

Figures

Figures reproduced from arXiv: 2507.10287 by the authors.

Figure 1
Figure 1. FIG. 1. Table of comparison between standard Variational [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Spin structure factor [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Energy density [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]

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Reference graph

Works this paper leans on

68 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [42]

    Park and M

    C.-Y. Park and M. J. Kastoryano, Phys. Rev. Res. 2, 023232 (2020)

    work page 2020

  2. [1]

    Promoting Scalars to Matrices The promotion of scalars to matrices is formulated functionally, such that a defined quantum value as scalar mapping of states ϕ → f (ϕ) is extended to analogous ma- trix mapping of bases Φ → F (Φ). The scalar mapping of states, which inherently is a mapping of rays, is invariant under multiplication by a constant |ϕ⟩ →x|ϕ⟩ t...

    work page

  3. [2]

    principal values

    Principal Values and Bases The construction of the matrix analog for a given quan- tum value ensures that the matrix eigenvalues are a set of N corresponding quantum values, which we will refer to as the “principal values”. Each principal value λi is the given quantum value or a similarly defined value for a corresponding state ui ∈ V. Collectively, these...

    work page

  4. [3]

    Operator Expectation Matrices We will refer to the matrix promotion of a pure state operator expectation value as an operator expectation 4 matrix (OEM). For a Hermitian operator ˆA and a basis Φ of V, the OEM eA(Φ) is given by the operator overlap matrix A(Φ) = [ [Φ| ˆA|Φ] ] normalized on the left by the in- verse of the Gram matrix, namely eA(Φ) = G−1(Φ...

    work page

  5. [4]

    For a Hermitian operator ˆA and basis Φ of V, the OVM corresponds to eΣ(A)(Φ) = eA(2)(Φ) − eA2(Φ), where eA(2) denotes the OEM for ˆA2 and eA2 = eA · eA

    Operator Variance and Covariance Matrices The corresponding operator variance matrices (OVMs) as the promotions of pure state operator variances are formed as matrix analogs of their scalar counterparts. For a Hermitian operator ˆA and basis Φ of V, the OVM corresponds to eΣ(A)(Φ) = eA(2)(Φ) − eA2(Φ), where eA(2) denotes the OEM for ˆA2 and eA2 = eA · eA....

    work page

  6. [5]

    determinant sampling scheme

    Monte Carlo Estimations of Operator Matrices The standard variational Monte Carlo (VMC) method for estimating operator values for a single quantum state can be extended to the estimations of their matrix pro- motions using the same Grassmann formulation, which we will refer to as Grassmann variational Monte Carlo (GVMC). Fig. 1 shows a table of comparison...

    work page

  7. [6]

    These can then be ob- tained from the operator matrices as their traces nor- malized by 1 /N

    Scalar Grassmann Operator Values Scalar Grassmann operator values can be defined nat- urally for each linear subspace V as the arithmetic means of the principal operator values. These can then be ob- tained from the operator matrices as their traces nor- malized by 1 /N. Without the normalization by 1 /N, these are all equivalently the corresponding pure ...

    work page

  8. [7]

    Its corresponding un- normalized amplitudes are given by: ⟨⟨S|Φ⟩⟩ = det [ [S|Φ] ]

    Wedge-Product States as Free Fermion States We remark that a linear subspace in the wedge- product representation can be interpreted as a state of N free fermions where single particle states are replaced by many-body wave functions. Its corresponding un- normalized amplitudes are given by: ⟨⟨S|Φ⟩⟩ = det [ [S|Φ] ]. (16) One can then define an extended Ham...

    work page

Show all 68 references
  1. [8]

    Fidelity Matrices The fidelity between two quantum states F (ϕ, ψ) = ⟨ϕ|ψ⟩ ⟨ψ|ϕ⟩ / ⟨ϕ|ϕ⟩ ⟨ψ|ψ⟩ is extended to the case of two bases Φ of V and Ψ of V ′ by promoting scalar overlaps to overlap matrices, following the same procedure used in the previous generalizations. The only...

  2. [9]

    variance

    Monte Carlo Estimations of Overlap and Fidelity Matrices Here we introduce a VMC method for estimating nor- malized overlap and fidelity matrices. Like for operator 7 value matrices, we use the same determinant sampling scheme Eq. (11) and formulate the sampling expressions as...

  3. [10]

    Goldilocks

    Metric Angle and Scalar Fidelity Grassmann metrics can be defined analogously to the Fubini-Study metric for rays as sign-less angles between linear subspaces. The metric angles are constructed nat- urally as some averages of the individual principal an- gles, as Fubini-Study ...

  4. [11]

    This is because the infinitesimal form metric angles α(p), from which the corresponding metric tensor is derived, reduce to the same values for all powers p

    Metric Tensor and Tangent Spaces Though there are multiple definitions of Grassmann metric angles, there is a singular standard definition of the Grassmann metric tensor which is ubiquitous in the literature. This is because the infinitesimal form metric angles α(p), from whic...

  5. [12]

    D. Pfau, S. Axelrod, H. Sutterud, I. von Glehn, and J. S. Spencer, Science 385, eadn0137 (2024)

    2024

  6. [13]

    J. F. Stanton and R. J. Bartlett, The Journal of Chemical Physics 98, 7029 (1993)

    1993

  7. [14]

    Troyer and U.-J

    M. Troyer and U.-J. Wiese, Phys. Rev. Lett. 94, 170201 (2005)

    2005

  8. [15]

    S. R. White, Phys. Rev. Lett. 69, 2863 (1992)

    1992

  9. [16]

    Verstraete and J

    F. Verstraete and J. I. Cirac, arXiv preprint (2004), arXiv:cond-mat/0407066

    2004 arXiv

  10. [17]

    Vidal, Phys

    G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)

    2007

  11. [18]

    Carleo and M

    G. Carleo and M. Troyer, Science 355, 602 (2017)

    2017

  12. [19]

    K. Choo, T. Neupert, and G. Carleo, Phys. Rev. B 100, 125124 (2019)

    2019

  13. [20]

    L. L. Viteritti, R. Rende, and F. Becca, Phys. Rev. Lett. 13 130, 236401 (2023)

    2023

  14. [21]

    L. L. Viteritti, R. Rende, A. Parola, S. Goldt, and F. Becca, Phys. Rev. B 111, 134411 (2025)

    2025

  15. [22]

    Chen and M

    A. Chen and M. Heyl, Nature Physics 20, 1476 (2024)

    2024

  16. [23]

    Denis and G

    Z. Denis and G. Carleo, Quantum 9, 1772 (2025)

    2025

  17. [24]

    K. Choo, A. Mezzacapo, and G. Carleo, Nature Commu- nications 11, 10.1038/s41467-020-15724-9 (2020)

    2020 doi

  18. [25]

    D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes, Phys. Rev. Research2, 033429 (2020)

    2020

  19. [26]

    Nomura, A

    Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada, Phys. Rev. B 96, 205152 (2017)

    2017

  20. [27]

    Stokes, J

    J. Stokes, J. R. Moreno, E. A. Pnevmatikakis, and G. Carleo, Phys. Rev. B 102, 205122 (2020)

    2020

  21. [28]

    Nys and G

    J. Nys and G. Carleo, Quantum 6, 833 (2022)

    2022

  22. [29]

    Schmitt and M

    M. Schmitt and M. Heyl, Phys. Rev. Lett. 125, 100503 (2020)

    2020

  23. [30]

    Sinibaldi, C

    A. Sinibaldi, C. Giuliani, G. Carleo, and F. Vicentini, Quantum 7, 1131 (2023)

    2023

  24. [31]

    J. Nys, G. Pescia, A. Sinibaldi, and G. Carleo, Nature Communications 15, 9404 (2024)

    2024

  25. [32]

    Sinibaldi, D

    A. Sinibaldi, D. Hendry, F. Vicentini, and G. Carleo, arXiv preprint (2024), arXiv:2412.11778

    2024 arXiv

  26. [33]

    K. Choo, G. Carleo, N. Regnault, and T. Neupert, Phys. Rev. Lett. 121, 167204 (2018)

    2018

  27. [34]

    M. T. Entwistle, Z. Sch¨ atzle, P. A. Erdman, J. Hermann, and F. No´ e, Nature Communications14, 274 (2023)

    2023

  28. [35]

    Pathak, B

    S. Pathak, B. Busemeyer, J. N. Rodrigues, and L. K. Wagner, The Journal of Chemical Physics 154 (2021)

    2021

  29. [36]

    W. L. McMillan, Phys. Rev. 138, A442 (1965)

    1965

  30. [37]

    Sorella, Phys

    S. Sorella, Phys. Rev. Lett. 80, 4558 (1998)

    1998

  31. [38]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre, Acta Appli- candae Mathematica 80, 199 (2004)

    2004

  32. [39]

    Hackl, T

    L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and J. I. Cirac, SciPost Phys. 9, 048 (2020)

    2020

  33. [40]

    Luo and B

    D. Luo and B. K. Clark, Phys. Rev. Lett. 122, 226401 (2019)

    2019

  34. [41]

    Stokes, J

    J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum 4, 269 (2020)

    2020

  35. [43]

    Z. Liu, H. Mao, C.-Y. Wu, C. Feichtenhofer, T. Darrell, and S. Xie, in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (2022) pp. 11976–11986

    2022

  36. [44]

    S. Woo, S. Debnath, R. Hu, X. Chen, Z. Liu, I. S. Kweon, and S. Xie, in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (2023) pp. 16133–16142

    2023

  37. [45]

    Sinibaldi, A

    A. Sinibaldi, A. F. Mello, M. Collura, and G. Carleo, arXiv preprint (2025), arXiv:2502.09725

    2025

  38. [46]

    R. P. Nutakki, A. Shokry, and F. Vicentini, arXiv preprint (2025), arXiv:2505.03466

    2025

  39. [47]

    J. L. Ba, J. R. Kiros, and G. E. Hinton, arXiv preprint (2016), arXiv:1607.06450

    2016 arXiv

  40. [48]

    Bengio, A

    Y. Bengio, A. Courville, and P. Vincent, IEEE transac- tions on pattern analysis and machine intelligence 35, 1798 (2013)

    2013

  41. [49]

    Denis and G

    Z. Denis and G. Carleo, arXiv preprint (2024), arXiv:2404.07869

    2024 arXiv

  42. [50]

    Rende, L

    R. Rende, L. L. Viteritti, L. Bardone, F. Becca, and S. Goldt, Communications Physics 7, 260 (2024)

    2024

  43. [51]

    Marshall, Proceedings of the Royal Society of London

    W. Marshall, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 232, 48 (1955)

    1955

  44. [52]

    A. W. Sandvik, Phys. Rev. B 56, 11678 (1997)

    1997

  45. [53]

    D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X. Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allah, et al. , Science 386, 296 (2024)

    2024

  46. [54]

    Goldshlager, N

    G. Goldshlager, N. Abrahamsen, and L. Lin, Journal of Computational Physics 516, 113351 (2024)

    2024

  47. [55]

    Motta, W

    M. Motta, W. Kirby, I. Liepuoniute, K. J. Sung, J. Cohn, A. Mezzacapo, K. Klymko, N. Nguyen, N. Yoshioka, and J. E. Rice, Electronic Structure 6, 013001 (2024)

    2024

  48. [56]

    I. L. Guti´ errez and C. B. Mendl, Quantum6, 627 (2022)

    2022

  49. [57]

    J´ onsson, B

    B. J´ onsson, B. Bauer, and G. Carleo, arXiv preprint (2018), arXiv:1808.05232

    2018 arXiv

  50. [58]

    Medvidovi´ c and G

    M. Medvidovi´ c and G. Carleo, npj Quantum Information 7, 1 (2021)

    2021

  51. [59]

    Donatella, Z

    K. Donatella, Z. Denis, A. Le Boit´ e, and C. Ciuti, Phys. Rev. A 108, 022210 (2023)

    2023

  52. [60]

    Gravina, V

    L. Gravina, V. Savona, and F. Vicentini, arXiv preprint (2024), arXiv:2410.10720

    2024 arXiv

  53. [61]

    Carleo et al

    G. Carleo et al. , SoftwareX , 100311 (2019)

    2019

  54. [62]

    Vicentini et al

    F. Vicentini et al. , SciPost Phys. Codebases , 7 (2022)

    2022

  55. [63]

    Westerhout, SpinED (2020)

    T. Westerhout, SpinED (2020)

    2020

  56. [64]

    Y. Ma, C. Liu, W. Li, S.-Y. Zhang, L.-M. Duan, Y. Wu, and D.-L. Deng, arXiv preprint (2025), arXiv:2506.08594. APPENDIX

    2025 arXiv

  57. [65]

    These identities describe how to compute matrix minors, namely determinants of square submatrices, with their complementary minors of the inverse matrix

    Matrix Minors Here we report key linear algebra identities that are used in the next sections of the Appendix for deriving the determinant sampling averages. These identities describe how to compute matrix minors, namely determinants of square submatrices, with their complemen...

  58. [66]

    This includes the case of an expectation value when |a⟩ = ˆA |ϕ⟩, for a given operator ˆA

    Determinant Sampling Averages: One Local Matrix Let us consider a general normalized overlap of the type ⟨ϕ|a⟩ / ⟨ϕ|ϕ⟩. This includes the case of an expectation value when |a⟩ = ˆA |ϕ⟩, for a given operator ˆA. For single-state VMC, the derivation of the average of the local e...

  59. [67]

    For the most general case of two ordered sets of states A = ( a1,

    Determinant Sampling Averages: Two Local Matrices The averages for two local matrices can be expressed most clearly in terms of their sampling covariances. For the most general case of two ordered sets of states A = ( a1, . . . , aN ) and B = ( b1, . . . , bN ) with local matr...

  60. [68]

    Numerical values In the tables below we show the numerical values for the variational energies obtained in this work. (qx, qy, qsf) Data |E0⟩ |E1⟩ |E2⟩ |E3⟩ (0, 0, 0) E/Ns EED/Ns −0.678871(3) −0.67887215 −0.655053(4) −0.65506482 −0.603897(6) −0.60391248 −0.602881(7) −0.6029117...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.