Pith. sign in

REVIEW 3 major objections 3 minor 72 references

Unbiased Diffusion Monte Carlo for non local operators

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Auxiliary Walking removes the trial-wavefunction bias from Diffusion Monte Carlo estimates of non-local operators.

desk verdict New idea for fixing DMC's 1RDM bias, but the key step — how to build the auxiliary walker in continuous space — is missing, so the method is not yet realizable. read the letter →

arxiv 2607.22273 v1 pith:RMKIXGG2 submitted 2026-07-24 cond-mat.mtrl-sci physics.comp-ph

Carlos Rodriguez Perez , Valerio Olevano , Francesco Sottile , Vitaly Gorelov This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.comp-ph PACS 71.15.-m02.70.Ss
keywords DiffusionMonteCarlonon-localoperatorsone-bodyreduceddensitymatrixForwardWalkingAuxiliarymixedestimatorbiastrialwavefunctionHubbarddimer
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

Diffusion Monte Carlo (DMC) computes ground-state expectations from a mixed distribution, and the standard Forward Walking correction fixes the distribution but leaves a separate trial-wavefunction bias inside the estimator for non-local quantities like the one-body reduced density matrix (1RDM). The paper introduces Auxiliary Walking (AUX), which reweights each DMC walker by the descendant count of an auxiliary walker that differs only in one electron position. The paper shows that this extra weight exactly cancels the non-local part of the accumulator bias, so AUX yields a pure 1RDM in the limits of zero time step and infinite walkers. The claim is demonstrated exactly on the symmetric Hubbard dimer and is shown to outperform Forward Walking on the helium atom.

What carries the argument

The central object is the auxiliary walker R'_i = {r', r2, ..., rN}, obtained from a walker R_i = {r1, r2, ..., rN} by changing one electron coordinate. Its asymptotic descendant count d(R'_i) is proportional to ψ0(R'_i)/ψT(R'_i), the same ratio that appears in the non-local part of the 1RDM accumulator. Reweighting each walker's mixed estimator O_M(R_i) by d(R'_i) produces the pure estimate O_0. The method is an extension of the Forward Walking principle, differing only in which walker's descendant count is used as the weight.

What would settle it

In a system with a known exact ground state, sample walkers from ψ_Tψ_0 and construct auxiliary walkers by moving one electron to a position drawn from ψ_0(R')/ψ_T(R'), computing their descendant counts analytically. If Σ d(R'_i) O_M(R_i) does not converge to O_0 as the number of walkers grows, the identity in Eq. (8) is false; the Hubbard dimer test is a discrete-state instance of this check, and the same test in a continuous system would settle the general claim.

Watch

Extended reading notes

Core claim

The central claim is that the equality O_A = Σ_i d(R'_i) O_M(R_i) = O_0 holds: reweighting each DMC walker R_i by the descendant number d(R'_i) of an auxiliary walker R'_i that differs from R_i only in one electron coordinate yields an unbiased pure expectation of a non-local operator. Forward Walking uses d(R_i) to correct only the sampling distribution, but the 1RDM estimator still contains ratios of trial wavefunctions in its accumulator. The auxiliary weight d(R'_i) is proportional to ψ0(R'_i)/ψT(R'_i), which is exactly the factor needed to remove the numerator bias, while the inverse weight d(R_i) cancels the denominator bias. Thus AUX removes both the mixed-distribution bias and the ac

Load-bearing premise

For every walker, an auxiliary walker differing in exactly one electron coordinate must exist among the sampled walker population so its descendant count can be measured; the paper asserts such a walker 'could be found or created' but provides no construction algorithm for continuous systems.

Editorial extensions

If this is right

  • AUX provides, for the first time within DMC, a path to unbiased 1RDMs in the zero-time-step and infinite-walker limit, removing a bias that Forward Walking cannot address.
  • The same reweighting scheme extends directly to other non-local observables, such as the momentum distribution and the two-body reduced density matrix, whose estimators also contain explicit wavefunction ratios.
  • In the Hubbard dimer, AUX recovers the exact pure 1RDM for all correlation strengths, while Forward Walking fails on the off-diagonal element.
  • In the helium atom, AUX outperforms Forward Walking at the same time step, showing that the accumulator bias is the dominant error for non-local operators once the distribution is corrected.

Reading between the lines

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

  • If the auxiliary-walker construction can be made practical in continuous space, AUX could replace Forward Walking as the standard DMC estimator for 1RDM-dependent quantities, such as natural orbitals and occupation numbers used in density-functional corrections.
  • The slow time-step convergence of the ψ_T cancellation observed in the helium atom suggests that the method's practical accuracy in realistic systems will hinge on reducing time-step errors or on using improved sampling schemes, not on the reweighting identity itself.
  • A possible testable extension is to apply AUX to systems with stronger correlation or with nodes, to see whether the fixed-node approximation interacts differently with the auxiliary-walker reweighting than it does with Forward Walking.
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 proposes 'Auxiliary Walking' (AUX), a reweighting scheme intended to remove the trial-wavefunction (accumulator) bias from Diffusion Monte Carlo estimates of non-local operators such as the one-body reduced density matrix. The central formula, Eq. (8), reweights each DMC walker R_i by the descendant number d(R'_i) of an auxiliary configuration R'_i that differs from R_i only in one electron coordinate. The authors claim that this reweighting yields the pure (ground-state) expectation value exactly. They benchmark AUX on a symmetric Hubbard dimer, where they report exact 1RDMs, and on the helium atom, where a single-time-step AUX result improves on forward walking but is not exact. The paper also argues that forward walking and the SUB/DIV corrections cannot remove the accumulator bias for non-local operators.

Significance. If Eq. (8) were correct, the paper would address a recognized limitation of DMC: obtaining unbiased, pure estimates of non-local observables without leaving the DMC framework. The problem is well motivated, and the algebraic structure of the proposed correction is appealing. The paper also ships reproducible code and a clean analytic dimer test, which are strengths. However, the central identity contains a normalization error in the Appendix A derivation, and the method as stated is not even formally exact. The additional lack of a construction for the auxiliary walker in continuous systems would already be a serious obstacle; the algebraic error makes the main claim untenable.

major comments (3)
  1. [Appendix A, Eqs. (A.10)-(A.11)] The step from the mixed-distribution average to the ψ_T^2 average omits the Radon-Nikodym normalization. For normalized averages, ⟨f⟩_{ψ_T ψ_0} = ⟨f(ψ_0/ψ_T)⟩_{ψ_T ψ_T} / ⟨ψ_0/ψ_T⟩_{ψ_T ψ_T}. Applying this to f = [ψ_T(R')/ψ_T(R)] δ d(R'_i), one obtains an extra denominator ⟨(1+Δ(R'))(1+Δ(R))⟩_{ψ_T^2}, whereas the exact ground-state 1RDM has denominator ⟨(1+Δ(R))^2⟩_{ψ_T^2}. These denominators are not equal in general, so Eq. (A.11) is not the exact ground-state 1RDM. A direct counterexample is N=1: R'_i is a fixed point r', d(R'_i) is constant, and Eq. (8) reduces to the mixed estimator, which is not O_0. The claimed exactness is therefore unsupported.
  2. [Sec. 2, Eq. (8); Sec. 3; Appendix C] Even setting the normalization issue aside, the paper does not specify how R'_i is found or created in a continuous system. In a continuum, the probability that a population walker lies exactly at R'_i is zero, and no auxiliary branching or interpolation algorithm is given. The Hubbard dimer avoids the issue because the configuration space is discrete; the helium AUX result is a single time step with no implementation details, and Appendix C documents only the dimer. Thus Eq. (8) is not a realizable method for the systems the paper targets.
  3. [Sec. 3, Fig. 4; Appendix C] The Hubbard dimer benchmark is constructed using the exact ψ_0 to sample the mixed distribution and to compute d(R)=ψ_0(R)/ψ_T(R) analytically. This tests an algebraic identity, not a stochastic DMC procedure for estimating descendant weights. It therefore cannot validate the practical AUX algorithm, and in light of the normalization error it does not constitute a proof of exactness. The statement that 'only AUX yields the exact 1RDM for all values of correlation' requires an implementation and a derivation that are not supplied.
minor comments (3)
  1. [Throughout] There are several typographical errors: 'derivate' (p. 8), 'pannel' (Fig. 1 caption), 'exptrapolation' (p. 11), 'roughtly' (p. 5), 'ammount' (p. 11), 'wihich' (p. 11).
  2. [Sec. 3, Fig. 3] The AUX result in Fig. 3 appears to be a single time step. The caption and text should state explicitly how this AUX calculation was performed for helium, including how the auxiliary walkers were obtained and how d(R'_i) was estimated.
  3. [Eq. (7)] The definition of γ_QMC is written as an unnormalized expectation; the subsequent normalization by γ(r';r') is mentioned in the text but should be stated in or near the equation for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (8) follows algebraically from the standard FWD descendant relation; the Hubbard dimer is a benchmark with known ψ0, not a fitted input.

full rationale

The central identity, Eq. (8), is derived, not assumed: Appendix A uses the externally established descendant relation d(R) ∝ ψ0(R)/ψT(R) (cited to Refs. [23] and [2]) and the definition ψ0 = ψT(1+Δ) to show that reweighting the mixed estimator by d(R'_i) cancels the trial-function factors in both the numerator and denominator of the 1RDM accumulator (Eqs. A.9–A.11). This is an algebraic reduction, not a redefinition of the target observable. The Hubbard dimer benchmark in Appendix C constructs the mixed distribution and d analytically from the known exact ψ0; that is a ground-truth consistency check, not a fitted parameter being passed off as a prediction. The paper does contain real gaps, but they are feasibility/validation gaps rather than circular steps: the text only says an auxiliary walker R'_i 'could be found (or created)' (Sec. 2, after Eq. (7)) and gives no algorithm or cost analysis for obtaining d(R'_i) in a continuous system; the only helium AUX point is a single time step with no implementation detail (Fig. 3, Appendix C). These affect the realizability and demonstrated accuracy of the method, not the logical independence of Eq. (8) from its inputs. Self-citations (Refs. [41], [42]) are used only for the helium exact solution and dimer exact solvability and are not load-bearing for the AUX claim. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central derivation rests on the standard FWD descendant property plus the new, unconstructed auxiliary walker. The Hubbard dimer benchmark injects exact ψ0 into both the sampled distribution and the weights, so it validates the algebra but not the practical method. The helium benchmark partially tests the method, but only at one time step and with a documented failure of the required error cancellation.

free parameters (2)
  • Trial wavefunction variational parameters for He = f_ep=0.777, w_ep=2.56, f_ee=0.41, w_ee=1.35, f_c=-0.5, w_c=0.1
    Appendix C gives optimized parameters in the VMC trial wavefunction; they define ψT and therefore control the size of the mixed-estimator and accumulator biases, though they are not the target of the derivation.
  • Simulation parameters: τ·N_w, forward time, number of steps = τ·N_w = 3000; forward time = 5 Ha; 40k steps
    Chosen by hand for convergence. AUX exactness is claimed only in the τ→0 and N_w→∞ limits, so these practical tunables are part of what is left uncontrolled in the helium demonstration.
assumptions (6)
  • domain assumption Forward-walking descendant property: d(R) ∝ ψ0(R)/ψT(R) in the asymptotic branching limit
    Used in Eq. (A.9) to replace d(R') with [1+Δ(R')] and d(R) with [1+Δ(R)]. This is a standard statistical property of FWD, but it is asymptotic and subject to population-control and time-step errors.
  • ad hoc to paper An auxiliary walker R'_i differing from R_i in exactly one electron coordinate can be found or created for every walker, with a well-defined descendant number
    The entire AUX estimator Eq. (8) depends on this. No algorithm is given for continuous systems; the Hubbard dimer implementation uses the exact ψ0 to compute d analytically, which is an internal construction rather than a general method.
  • domain assumption The walker population exactly samples the mixed distribution ψ*T ψ0
    AUX exactness in Eq. (A.11) uses [1+Δ(R)] from the mixed distribution. Appendix B documents that this is violated in practice and that the ψT cancellation converges slowly with time step.
  • domain assumption The trial wavefunction is nodeless (strictly positive) in both benchmarks
    Both the helium atom and the Hubbard dimer ground states are nodeless, so fixed-node bias is absent. The paper does not address how AUX behaves in systems with nodal surfaces.
  • standard math Ground-state and trial wavefunctions are real
    The derivations write ψ0 and ψT without complex conjugation in ratios; this is valid for the non-degenerate real ground states of the benchmarks but is an assumption for more general systems.
  • domain assumption For the SUB/DIV correction estimates, ψ0 = ψT(1+Δ) with Δ small
    Appendix A truncates at linear order in Δ to derive the subtraction and division formulas. This is not central to AUX, but it is part of the paper's comparison framework.
invented entities (1)
  • Auxiliary walker R'_i
    purpose: To supply a descendant reweighting d(R'_i) at a displaced configuration so the non-local part of the 1RDM accumulator is unbiased while the distribution part is corrected by d(R_i).
    No construction is given in continuous space. In the Hubbard dimer it is realized by using the known exact ψ0 to compute d analytically, which is an internal validation rather than independent evidence of the entity's utility.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Unbiased Diffusion Monte Carlo for non local operators." pith.science (2026). https://pith.science/paper/RMKIXGG2

@misc{pith2026260722273,
  author       = {Pith},
  title        = {Pith review of: Unbiased Diffusion Monte Carlo for non local operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMKIXGG2}},
  note         = {Machine review of arXiv:2607.22273}
}
read the original abstract

We propose a new mathematically exact method for computing unbiased Diffusion Monte Carlo (DMC) estimates of non-local operators. We demonstrate that the current state-of- the-art technique, Forward Walking, is only exact for local quantities and fails to yield unbiased results for the non-local components of reduced density matrices (RDMs). Our method significantly outperforms Forward Walking, as shown in two systems: in the symmetric Hubbard dimer it yields a pure 1RDM; while in the Helium atom it will give an unbiased 1RDM in the limits of zero time step and infinite walkers.

Figures

Figures reproduced from arXiv: 2607.22273 by the authors.

Figure 1
Figure 1. Schematic representation of FWD (left) and AUX (right). Left figure rep [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. QMC results for the density, we show VMC (grey), DMC (purple), SUB [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. QMC results for the 1RDM along the line γ (1) (X, 0, 0; 0.5, 0, 0), we show VMC (grey), DMC (purple), SUB correction (orange), FWD (blue) and AUX (red). a) Real space 1RDM deviation from exact result (at a time step of τ = 0.005). b) Root Mean Square Error (RMSE) of the 1RDM along the previous line. The symmetric Hubbard dimer with two opposite spin electrons is exactly solvable [42] and can shed light on the nature… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Density and off diagonal element of the 1RDM of the symmetric Hubbard [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: , where we compute the ratio of the QMC A(r) to the exact one. A naive approach to removing the ψT error would be to use FWD, as it will lead to sampling with ψ0ψ0 . This will not work, as using FWD will remove the possibility for error cancellation. Using DMC should a…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 9 canonical work pages

  1. [1]

    H. A. Bethe, Zur Theorie der Metalle. i. Eigenwerte und Eigenfunktionen der linearen Atomkette , Zeit. f \"u r Phys. 71 , 205 (1931), 10.1007\

    1931

  2. [2]

    Ginsparg, It was twenty years ago today

    P. Ginsparg, It was twenty years ago today... , http://arxiv.org/abs/1108.2700

    arXiv

  3. [3]

    Foulkes, W. M. C. and Mitas, L. and Needs, R. J. and Rajagopal, G. , journal =. 2001 , month =. doi:10.1103/RevModPhys.73.33 , url =

    doi:10.1103/revmodphys.73.33 2001

  4. [4]

    W. K. Hastings , journal=. 1970 , volume=

    1970

  5. [5]

    Communications on Pure and Applied Mathematics , volume =

    Kato, Tosio , title =. Communications on Pure and Applied Mathematics , volume =. doi:https://doi.org/10.1002/cpa.3160100201 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.3160100201 , year =

    doi:10.1002/cpa.3160100201

  6. [6]

    Ceperley and Lucia Reining and Richard M

    David M. Ceperley and Lucia Reining and Richard M. Martin , title =. 2016 , address =. doi:https://doi.org/10.1017/CBO9781139050807 , isbn =

    doi:10.1017/cbo9781139050807 2016

  7. [7]

    1929 , month =

    Hylleraas, E.A , journal =. 1929 , month =. doi:https://doi.org/10.1007/BF01375457 , url =

    work page doi:10.1007/bf01375457 1929

  8. [9]

    and Rosenbluth, Marshall N

    Metropolis, Nicholas and Rosenbluth, Arianna W. and Rosenbluth, Marshall N. and Teller, Augusta H. and Teller, Edward , title = ". J. Chem. Phys. , volume =. 1953 , month =. doi:https://doi.org/10.1063/1.1699114 , doi =

    doi:10.1063/1.1699114 1953

Show all 72 references
  1. [10]

    2008 , doi =

    Trial wave functions for high-pressure metallic hydrogen , journal =. 2008 , doi =

    2008

  2. [11]

    Inhomogeneous Electron Gas , author =. Phys. Rev. , volume =. 1964 , month =. doi:10.1103/PhysRev.136.B864 , url =

    1964 doi

  3. [12]

    Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study , author =. Phys. Rev. B , volume =. 1998 , month =. doi:10.1103/PhysRevB.58.6800 , url =

    1998 doi

  4. [13]

    Ragot, Sébastien and Ruiz, María Belén , title = ". J. Chem. Phys. , volume =. 2008 , doi =

    2008

  5. [14]

    Improved Monte Carlo estimators for the one-body density , author =. Phys. Rev. E , volume =. 2007 , month =. doi:10.1103/PhysRevE.75.035701 , url =

    2007 doi

  6. [15]

    Ab initio electronic density in solids by many-body plane-wave auxiliary-field quantum Monte Carlo calculations , author =. Phys. Rev. B , volume =. 2021 , month =. doi:10.1103/PhysRevB.103.075138 , url =

    2021 doi

  7. [16]

    Highly accurate real-space electron densities with neural networks , author=. J. Chem. Phys. , volume=. doi:https://doi.org/10.1063/5.0236919 , year=

    doi

  8. [17]

    Static Response and Local Field Factor of the Electron Gas , author =. Phys. Rev. Lett. , volume =. 1995 , month =. doi:10.1103/PhysRevLett.75.689 , url =

    1995 doi

  9. [18]

    Accurate and simple analytic representation of the electron-gas correlation energy , author =. Phys. Rev. B , volume =. 1992 , month =. doi:10.1103/PhysRevB.45.13244 , url =

    1992 doi

  10. [19]

    Accurate

    Aouina, Ayoub and Gatti, Matteo and Chen, Siyuan and Zhang, Shiwei and Reining, Lucia , journal =. Accurate. 2023 , month =. doi:10.1103/PhysRevB.107.195123 , url =

    2023 doi

  11. [20]

    From electron densities to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies , author =. Phys. Rev. A , volume =. 1994 , month =. doi:10.1103/PhysRevA.50.2138 , url =

    1994 doi

  12. [21]

    Medvedev and Ivan S

    Michael G. Medvedev and Ivan S. Bushmarinov and Jianwei Sun and John P. Perdew and Konstantin A. Lyssenko , title =. Science , volume =. 2017 , doi =

    2017

  13. [22]

    Varsano, Daniele and Barborini, Matteo and Guidoni, Leonardo , title =. J. Chem. Phys. , volume =. 2014 , month =. doi:10.1063/1.4863213 , url =

    2014 doi

  14. [23]

    Flaviano Della Pia and Benjamin Shi and Yasmine S. Al-Hamdani and Dario Alfè and Tyler Anderson and Matteo Barborini and Anouar Benali and Michele Casula and Neil Drummond and Matúš Dubecký and Claudia Filippi and Paul Kent and Jaron Krogel and Pablo Lopez Rios and Arne Lüchow...

    arXiv

  15. [24]

    and Pathak, Shivesh and Kleiner, Kevin G

    Wheeler, William A. and Pathak, Shivesh and Kleiner, Kevin G. and Yuan, Shunyue and Rodrigues, João N. B. and Lorsung, Cooper and Krongchon, Kittithat and Chang, Yueqing and Zhou, Yiqing and Busemeyer, Brian and Williams, Kiel T. and Muñoz, Alexander and Chow, Chun Yu and Wagn...

    2023 doi

  16. [25]

    Accurate electron densities from quantum Monte Carlo calculations using real-space grids , author=. J. Chem. Phys. , volume=. 2025 , doi =

    2025

  17. [26]

    Toward Large-Scale AFQMC Calculations: Large Time Step Auxiliary-Field Quantum Monte Carlo , author=. J. Chem. Theory Comput. , volume=. 2024 , doi =

    2024

  18. [27]

    Backflow correlations for the electron gas and metallic hydrogen , author=. Phys. Rev. E , volume=. 2003 , doi =

    2003

  19. [28]

    Quantum hard spheres in a channel , author=. Phys. Rev. A , volume=. 1974 , doi =

    1974

  20. [29]

    Systematic study of finite-size effects in quantum Monte Carlo calculations of real metallic systems , author=. J. Chem. Phys. , volume=. 2015 , doi =

    2015

  21. [30]

    Finite-size correction in many-body electronic structure calculations , author=. Phys. Rev. Lett. , volume=. 2008 , doi =

    2008

  22. [31]

    Renormalization factor and effective mass of the two-dimensional electron gas , author =. Phys. Rev. B , volume =. 2009 , month =. doi:10.1103/PhysRevB.79.041308 , url =

    2009 doi

  23. [32]

    Wuttig, Matthias and Schön, Carl-Friedrich and Lötfering, Jakob and Golub, Pavlo and Gatti, Carlo and Raty, Jean-Yves , title =. Adv. Mater. , volume =. doi:https://doi.org/10.1002/adma.202208485 , year =

    doi

  24. [33]

    Electron localization properties in high pressure hydrogen at the liquid-liquid phase transition by

    Pierleoni, Carlo and Rillo, Giovanni and Ceperley, David M and Holzmann, Markus , booktitle=. Electron localization properties in high pressure hydrogen at the liquid-liquid phase transition by. 2018 , doi =

    2018

  25. [34]

    A random-walk simulation of the Schr

    Anderson, James B , journal=. A random-walk simulation of the Schr. 1975 , doi =

    1975

  26. [35]

    Energy of a boson fluid with

    Kalos, MH , journal=. Energy of a boson fluid with. 1970 , doi =

    1970

  27. [36]

    Ground state of the electron gas by a stochastic method , author=. Phys. Rev. Lett. , volume=. 1980 , doi =

    1980

  28. [37]

    Ground state of a homogeneous Bose gas: A diffusion Monte Carlo calculation , author=. Phys. Rev. A , volume=. doi:https://doi.org/10.1103/PhysRevA.60.5129 , year=

    doi

  29. [38]

    Electronic band gaps from quantum

    Yang, Yubo and Gorelov, Vitaly and Pierleoni, Carlo and Ceperley, David M and Holzmann, Markus , journal=. Electronic band gaps from quantum. 2020 , doi =

    2020

  30. [39]

    Calculations of silicon self-interstitial defects , author=. Phys. Rev. Lett. , volume=. 1999 , doi =

    1999

  31. [40]

    Condensate fraction in liquid

    Moroni, S and Boninsegni, M , journal=. Condensate fraction in liquid. 2004 , doi =

    2004

  32. [41]

    Reptation quantum

    Carleo, Giuseppe and Becca, Federico and Moroni, Saverio and Baroni, Stefano , journal=. Reptation quantum. 2010 , doi =

    2010

  33. [42]

    Fixed-node diffusion

    Sottile, F and Ballone, P , journal=. Fixed-node diffusion. 2001 , doi =

    2001

  34. [43]

    Diffusion monte carlo study of atomic systems from

    Hongo, Kenta and Kawazoe, Yoshiyuki and Yasuhara, Hiroshi , journal=. Diffusion monte carlo study of atomic systems from. 2006 , doi =

    2006

  35. [44]

    A diffusion

    Umrigar, Cyrus J and Nightingale, MP and Runge, KJ , journal=. A diffusion. 1993 , doi =

    1993

  36. [45]

    Reducing the time-step errors in diffusion

    Anderson, Tyler A and Per, Manolo C and Umrigar, CJ , journal=. Reducing the time-step errors in diffusion. 2024 , doi =

    2024

  37. [46]

    1994 , doi =

    Kwon, Yongkyung and Ceperley, DM and Martin, Richard M , journal=. 1994 , doi =

    1994

  38. [47]

    Inhomogeneous backflow transformations in quantum

    L. Inhomogeneous backflow transformations in quantum. Phys. Rev. E , volume=. 2006 , doi =

    2006

  39. [48]

    Twist-averaged boundary conditions in continuum quantum

    Lin, C and Zong, FH and Ceperley, David M , journal=. Twist-averaged boundary conditions in continuum quantum. 2001 , doi =

    2001

  40. [49]

    Finite-size error in many-body simulations with long-range interactions , author=. Phys. Rev. Lett. , volume=. 2006 , doi =

    2006

  41. [50]

    Finite-size errors in continuum quantum

    Drummond, ND and Needs, RJ and Sorouri, A and Foulkes, WMC , journal=. Finite-size errors in continuum quantum. 2008 , doi =

    2008

  42. [51]

    Reptation quantum

    Baroni, Stefano and Moroni, Saverio , journal=. Reptation quantum. 1999 , doi =

    1999

  43. [52]

    Zhang, Shiwei and Krakauer, Henry , journal=. Quantum. 2003 , doi =

    2003

  44. [53]

    A random-walk simulation of the

    Anderson, James B , journal=. A random-walk simulation of the. 1975 , doi =

    1975

  45. [54]

    Beyond the locality approximation in the standard diffusion Monte Carlo method , author=. Phys. Rev. B—Condensed Matter and Materials Physics , volume=. 2006 , doi =

    2006

  46. [55]

    Monte Carlo methods in statistical physics , pages=

    Quantum many-body problems , author=. Monte Carlo methods in statistical physics , pages=. 1979 , doi =

    1979

  47. [56]

    Ground state of liquid

    McMillan, William Lauchlin , journal=. Ground state of liquid. 1965 , doi =

    1965

  48. [57]

    Helium at zero temperature with hard-sphere and other forces , author=. Phys. Rev. A , volume=. 1974 , doi =

    1974

  49. [58]

    2020 , doi =

    Yang, Yubo and Hiraoka, Nozomu and Matsuda, Kazuhiro and Holzmann, Markus and Ceperley, David M , journal=. 2020 , doi =

    2020

  50. [59]

    1999 , doi =

    Filippi, Claudia and Ceperley, David M , journal=. 1999 , doi =

    1999

  51. [60]

    Benchmark all-electron ab initio quantum

    Nemec, Norbert and Towler, Michael D and Needs, RJ , journal=. Benchmark all-electron ab initio quantum. 2010 , doi =

    2010

  52. [61]

    Theory of finite size effects for electronic quantum

    Holzmann, Markus and Clay III, Raymond C and Morales, Miguel A and Tubman, Norm M and Ceperley, David M and Pierleoni, Carlo , journal=. Theory of finite size effects for electronic quantum. 2016 , doi =

    2016

  53. [62]

    Bilinear quantum

    Zhang, Shiwei and Kalos, Malvin H , journal=. Bilinear quantum. 1993 , doi =

    1993

  54. [63]

    Numerical study of the two-dimensional

    Buonaura, Matteo Calandra and Sorella, Sandro , journal=. Numerical study of the two-dimensional. 1998 , doi =

    1998

  55. [64]

    2026 , doi =

    Ghaffar, Abdul and Zhang, Shenli and Wan, Liwen F and Saritas, Kayahan and Reboredo, Fernando A and Krogel, Jaron T , journal=. 2026 , doi =

    2026

  56. [65]

    A first-principles

    Wines, Daniel and Saritas, Kayahan and Ataca, Can , journal=. A first-principles. 2020 , doi =

    2020

  57. [66]

    Systematic

    Wines, Daniel and Choudhary, Kamal and Tavazza, Francesca , journal=. Systematic. 2023 , doi =

    2023

  58. [67]

    Unbiased estimators in quantum

    Casulleras, J and Boronat, J , journal=. Unbiased estimators in quantum. 1995 , doi =

    1995

  59. [68]

    Accurate electron densities from quantum

    Kaiser, Alexander and K. Accurate electron densities from quantum. J. Chem. Phys. , volume=. 2025 , doi =

    2025

  60. [69]

    Improved

    Assaraf, Roland and Caffarel, Michel and Scemama, Anthony , journal=. Improved. 2007 , doi =

    2007

  61. [70]

    1991 , issn =

    Monte Carlo algorithms for expectation values of coordinate operators , journal =. 1991 , issn =. doi:https://doi.org/10.1016/0021-9991(91)90236-E , url =

    1991 doi

  62. [71]

    Total energy beyond

    El-Sahili, Abdallah and Sottile, Francesco and Reining, Lucia , journal=. Total energy beyond. 2024 , doi =

    2024

  63. [72]

    GitHub repository , howpublished =

  64. [73]

    Li, Jing and Drummond, N. D. and Schuck, Peter and Olevano, Valerio , journal =. Comparing many-body approaches against the helium atom exact solution , pages =. 2019 , publisher =. doi:10.21468/SciPostPhys.6.4.040 , url =

    2019 doi

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.