pith. sign in

arxiv: 2606.08946 · v1 · pith:MDIW752Vnew · submitted 2026-06-08 · 📊 stat.CO · physics.chem-ph· physics.comp-ph

A Diffusion Monte Carlo algorithm employing depth first traversal and a stack instead of a swarm

Bastiaan J. Braams This is my paper

Pith reviewed 2026-06-27 14:26 UTC · model grok-4.3

classification 📊 stat.CO physics.chem-phphysics.comp-ph
keywords Diffusion Monte Carlodepth first traversalstack based algorithmpopulation controldescendant weightingparticle transportmemory efficiencyMonte Carlo methods
0
0 comments X

The pith

A depth-first stack-based Diffusion Monte Carlo algorithm is more memory efficient than the traditional walker swarm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a depth-first traversal method using a stack for Diffusion Monte Carlo simulations, called DMCD, instead of following a swarm of walkers. This change is presented as reducing overall memory use and improving fit with memory hierarchies and co-processors while handling population control and descendant weighting in a natural way. The approach unifies the algorithmic treatment of the DMC eigenvalue problem with the linear equation problem from particle transport. A pool of starter configurations is maintained to handle cases when the stack empties, and the paper states that this maintains unbiased results. A complete code implementation is provided to support the claims.

Core claim

The depth first approach, called DMCD here, can be more memory efficient than the breadth first approach, both for total memory and for use of a memory hierarchy and of co-processors. The implementation appears very natural for population control and for descendant weighting and it unifies algorithmic treatment of the eigenvalue problem (DMC) with the linear equation problem (particle transport). A concern with DMCD that is not present in the breadth first approach, and that is successfully addressed here, is the need to maintain a pool of starters for use when a new walker is required and the stack is empty.

What carries the argument

The depth-first traversal of configurations using a stack together with a maintained pool of starter walkers for Diffusion Monte Carlo.

If this is right

  • The method requires less total memory and works better with memory hierarchies and co-processors than swarm-based DMC.
  • Population control and descendant weighting follow directly from the stack structure without extra machinery.
  • The same code structure handles both the DMC eigenvalue problem and particle transport linear equations.
  • The starter pool resolves the main new concern of the depth-first method while preserving unbiased sampling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stack approach may scale more readily to very large configuration spaces where swarm memory would become prohibitive.
  • It could simplify hybrid codes that combine DMC with transport calculations on shared infrastructure.
  • Hardware with limited fast memory, such as GPUs or embedded systems, might see larger gains from the reduced footprint.

Load-bearing premise

Maintaining a pool of starter configurations when the stack is empty produces statistically unbiased results equivalent to the traditional swarm method without introducing significant overhead or bias.

What would settle it

Direct numerical comparison of ground-state energies or other observables computed by the stack-based DMCD code and by a standard swarm DMC code on an identical test problem, checking for agreement within statistical error bars.

read the original abstract

Diffusion Monte Carlo (DMC) and Monte Carlo for particle transport with importance sampling both involve simulations of weighted walkers that undergo birth and death processes (splitting and Russian Roulette). The established implementations of these methods are quite different: Particle simulation Monte Carlo employs a stack to handle the splitting history whereas in traditional DMC one follows a swarm of walkers. The particle simulation Monte Carlo approach involves a depth first traversal of the visited configurations whereas the traditional DMC approach may be seen as a breadth first traversal. In the present work the implementation of a depth first, stack based approach to DMC is described and a complete code is presented. The depth first approach, called DMCD here, can be more memory efficient than the breadth first approach, both for total memory and for use of a memory hierarchy and of co-processors. The implementation appears very natural for population control and for descendant weighting and it unifies algorithmic treatment of the eigenvalue problem (DMC) with the linear equation problem (particle transport). A concern with DMCD that is not present in the breadth first approach, and that is successfully addressed here, is the need to maintain a pool of starters for use when a new walker is required and the stack is empty. The DMCD approach appears to have the potential to become the preferred implementation for many DMC applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents DMCD, a depth-first traversal algorithm for Diffusion Monte Carlo that uses a stack in place of the conventional breadth-first swarm of walkers. It claims this yields improved memory efficiency (total and hierarchical), natural handling of population control and descendant weighting, algorithmic unification with particle-transport Monte Carlo, and resolution of the starter-pool issue when the stack empties, all while preserving statistical equivalence; a complete code implementation is supplied.

Significance. If the statistical equivalence holds, the approach could alter standard practice in DMC and related weighted-walker methods by reducing memory footprint and unifying eigenvalue and linear-transport problems. The provision of complete, reproducible code is a clear strength that supports verification and adoption.

major comments (2)
  1. [§3.2] §3.2 (starter-pool description): the central claim of unbiased equivalence requires that pool sampling when the stack empties preserves the joint distribution of position, weight, and age; no derivation or invariance argument is supplied showing that the depth-first insertion and sampling rules commute with the branching/diffusion steps, leaving the equivalence assertion without formal support.
  2. [§4] §4 (numerical results): the reported eigenvalue agreement with the swarm method is presented only for a single test case; without additional benchmarks or a direct comparison of the full population statistics (not just the eigenvalue), it is impossible to confirm that the starter-pool mechanism introduces no detectable bias or overhead.
minor comments (2)
  1. Notation for walker age and descendant weighting is introduced without an explicit table relating it to the corresponding swarm quantities; a small comparison table would improve clarity.
  2. The abstract states that the starter-pool concern 'is successfully addressed here' but the manuscript body does not contain a dedicated subsection summarizing the proof or test that establishes this; relocating or adding such a summary would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (starter-pool description): the central claim of unbiased equivalence requires that pool sampling when the stack empties preserves the joint distribution of position, weight, and age; no derivation or invariance argument is supplied showing that the depth-first insertion and sampling rules commute with the branching/diffusion steps, leaving the equivalence assertion without formal support.

    Authors: The starter-pool mechanism is constructed so that every walker inserted into the pool has already experienced the identical diffusion, branching, and weight-update steps as in the conventional swarm; sampling from the pool when the stack empties therefore draws from the same joint distribution by design. We nevertheless agree that an explicit invariance argument would strengthen the presentation and will add a concise derivation of the required commutation property to the revised §3.2. revision: yes

  2. Referee: [§4] §4 (numerical results): the reported eigenvalue agreement with the swarm method is presented only for a single test case; without additional benchmarks or a direct comparison of the full population statistics (not just the eigenvalue), it is impossible to confirm that the starter-pool mechanism introduces no detectable bias or overhead.

    Authors: The single benchmark was selected as a standard, well-characterized test; the complete code supplied with the manuscript permits readers to run further cases. We accept that expanding the numerical section would improve confidence and will therefore add at least one additional benchmark together with direct comparisons of the full population statistics (weight and age histograms) in the revised §4. revision: yes

Circularity Check

0 steps flagged

No circularity: direct algorithmic description with no fitted predictions or self-referential derivations

full rationale

The paper presents a stack-based depth-first implementation of Diffusion Monte Carlo (DMCD) as an alternative to the traditional swarm (breadth-first) method. No equations, parameters, or results are fitted to data and then re-predicted; the description is self-contained as a code-level algorithm with population control and starter-pool handling stated to preserve equivalence. No self-citations, uniqueness theorems, or ansatzes from prior author work are invoked as load-bearing. The equivalence claim for the starter pool is asserted without reducing to a definitional identity or statistical tautology within the provided text. This is a standard non-circular methodological contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper introduces no free parameters or invented entities and relies only on the standard domain assumptions of Monte Carlo methods with importance sampling and branching.

axioms (1)
  • domain assumption Standard Monte Carlo assumptions for importance sampling and birth-death processes remain valid when switching from breadth-first to depth-first traversal.
    The paper builds directly on the established DMC framework without altering its statistical foundations.

pith-pipeline@v0.9.1-grok · 5766 in / 1137 out tokens · 32037 ms · 2026-06-27T14:26:40.753692+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 1 linked inside Pith

  1. [1]

    Kahn, Herman.Applications of Monte Carlo. No. AECU-3259; RM-1237-AEC. RAND Corp., Santa Monica, CA (United States), 1954

    1954

  2. [2]

    Carter, Leland Lavele, and Edmond Darrell Cashwell.Particle-transport simulation with the Monte Carlo method. No. TID– 26607. Los Alamos Scientific Lab., N. Mex.(USA), 1974

    1974

  3. [3]

    MCNPTM-A general Monte Carlo N-particle transport code

    Briesmeister, Judith F. “MCNPTM-A general Monte Carlo N-particle transport code.” Version 4C, LA-13709-M, Los Alamos National Laboratory 2 (2000)

    2000

  4. [4]

    Monte Carlo Methods

    Spanier, Jerome. “Monte Carlo Methods.” InNuclear Computational Science: A Century in Review, pp. 117-165. Dordrecht: Springer Netherlands, 2009

    2009

  5. [5]

    The OpenMC Monte Carlo particle transport code

    Romano, Paul K., and Benoit Forget. “The OpenMC Monte Carlo particle transport code.”Annals of Nuclear Energy51 (2013): 274-281

    2013

  6. [6]

    CRC press, 2018

    Lux, Iv´ an, L´ aszl´ o Koblinger.Monte Carlo Particle Transport Methods. CRC press, 2018

    2018

  7. [7]

    Monte Carlo methods for the neutron transport equation

    Cox, Alexander MG, Simon C. Harris, Andreas E. Kyprianou, and Minmin Wang. “Monte Carlo methods for the neutron transport equation.”SIAM/ASA Journal on Uncertainty Quantification10, no. 2 (2022): 775-825

    2022

  8. [8]

    Oxford University Press, 2007

    Anderson, James B.Quantum Monte Carlo: origins, development, applications. Oxford University Press, 2007

    2007

  9. [9]

    Quantum Monte Carlo studies of vibrational states in molecules and clusters

    Suhm, Martin A., and Robert O. Watts. “Quantum Monte Carlo studies of vibrational states in molecules and clusters.” Physics Reports204, no. 4 (1991): 293-329

    1991

  10. [10]

    A diffusion Monte Carlo algorithm with very small time-step errors

    Umrigar, Cyrus J., M. P. Nightingale, and K. J. Runge. “A diffusion Monte Carlo algorithm with very small time-step errors.” The Journal of chemical physics 99, no. 4 (1993): 2865-2890

    1993

  11. [11]

    Peter, and Cyrus J

    Nightingale, M. Peter, and Cyrus J. Umrigar, eds.Quantum Monte Carlo methods in physics and chemistry. NATO ASI Series C: Mathematical and Physical Sciences, No. 525. Springer Science & Business Media, 1998

    1998

  12. [12]

    Quantum Monte Carlo simulations of solids

    Foulkes, William MC, Lubos Mitas, R. J. Needs, and Guna Rajagopal. “Quantum Monte Carlo simulations of solids.”Reviews of Modern Physics73, no. 1 (2001): 33

    2001

  13. [13]

    Towler, Neil D

    Needs, Richarad J., Michael D. Towler, Neil D. Drummond, and P. L´ opez R´ ıos. ”Continuum variational and diffusion quantum Monte Carlo calculations.” em Journal of Physics: Condensed Matter 22, no. 2 (2010): 023201

    2010

  14. [14]

    Quantum Monte Carlo and related approaches

    Austin, Brian M., Dmitry Yu Zubarev, and William A. Lester Jr. “Quantum Monte Carlo and related approaches.”Chemical Reviews112, no. 1 (2012): 263-288

    2012

  15. [15]

    A brief introduction to the diffusion Monte Carlo method and the fixed-node approximation

    Annarelli, Alfonso, Dario Alf` e, and Andrea Zen. “A brief introduction to the diffusion Monte Carlo method and the fixed-node approximation.”Journal of Chemical Physics161, no. 24 (2024)

    2024

  16. [16]

    Diffusion Monte Carlo approaches for studying nuclear quantum effects in fluxional molecules

    DiRisio, Ryan J., Jacob M. Finney, and Anne B. McCoy. “Diffusion Monte Carlo approaches for studying nuclear quantum effects in fluxional molecules.”Wiley Interdisciplinary Reviews: Computational Molecular Science12, no. 6 (2022): e1615

    2022

  17. [17]

    On Sequential Monte Carlo sampling methods for Bayesian filter- ing

    Doucet, Arnaud, Simon Godsill, and Christophe Andrieu. “On Sequential Monte Carlo sampling methods for Bayesian filter- ing.”Statistics and Computing10, no. 3 (2000): 197-208

    2000

  18. [18]

    Sequential Monte Carlo Samplers

    Del Moral, Pierre, Arnaud Doucet, and Ajay Jasra. “Sequential Monte Carlo Samplers.”Journal of the Royal Statistical Society Series B: Statistical Methodology68, no. 3 (2006): 411-436

    2006

  19. [19]

    An overview of existing methods and recent advances in Sequential Monte Carlo

    Capp´ e, Olivier, Simon J. Godsill, and Eric Moulines. “An overview of existing methods and recent advances in Sequential Monte Carlo.”Proceedings of the IEEE95, no. 5 (2007): 899-924. DIFFUSION MONTE CARLO ALGORITHM ... 11

    2007

  20. [20]

    Particle Markov Chain Monte carlo Methods

    Andrieu, Christophe, Arnaud Doucet, and Roman Holenstein. “Particle Markov Chain Monte carlo Methods.”Journal of the Royal Statistical Society Series B: Statistical Methodology72, no. 3 (2010): 269-342

    2010

  21. [21]

    Elements of Sequential Monte Carlo

    Naesseth, Christian A., Fredrik Lindsten, and Thomas B. Sch¨ on. “Elements of Sequential Monte Carlo.”Foundations and Trends in Machine Learning12, no. 3 (2019): 187-306

    2019

  22. [22]

    Springer, 2020

    Chopin, Nicolas, and Omiros Papaspiliopoulos.An Introduction to Sequential Monte Carlo. Springer, 2020

    2020

  23. [23]

    An Invitation to Sequential Monte Carlo Samplers

    Dai, Chenguang, Jeremy Heng, Pierre E. Jacob, and Nick Whiteley. “An Invitation to Sequential Monte Carlo Samplers.” Journal of the American Statistical Association117, no. 539 (2022): 1587-1600

    2022

  24. [24]

    Oates and Chris Sherlock.Scalable Monte Carlo for Bayesian Learning

    Paul Fearnhead, Christopher Nemeth, Chris J. Oates and Chris Sherlock.Scalable Monte Carlo for Bayesian Learning. Cambridge University Press, 2025

    2025

  25. [25]

    Unbiased and consistent nested sampling via sequential Monte Carlo

    Salomone, Robert, Leah F. South, Christopher Drovandi, Dirk P. Kroese, and Adam M. Johansen. “Unbiased and consistent nested sampling via sequential Monte Carlo.”Journal of the Royal Statistical Society Series B: Statistical Methodology87, no. 4 (2025): 1221-1238

    2025

  26. [26]

    Improved diffusion Monte Carlo

    Hairer, Martin, and Jonathan Weare. “Improved diffusion Monte Carlo.”Communications on Pure and Applied Mathematics 67, no. 12 (2014): 1995-2021

    2014

  27. [27]

    Fast randomized iteration: Diffusion Monte Carlo through the lens of numerical linear algebra

    Lim, Lek-Heng, and Jonathan Weare. “Fast randomized iteration: Diffusion Monte Carlo through the lens of numerical linear algebra.”SIAM Review59, no. 3 (2017): 547-587

    2017

  28. [28]

    Unifying sequential Monte Carlo with resampling matrices

    Webber, Robert J. “Unifying sequential Monte Carlo with resampling matrices.” arXiv preprint arXiv:1903.12583 (2019)

    Pith/arXiv arXiv 1903

  29. [29]

    Diffusion Monte Carlo methods with a fixed number of walkers

    Assaraf, Roland, Michel Caffarel, and Anatole Khelif. “Diffusion Monte Carlo methods with a fixed number of walkers.” Physical Review E61, no. 4 (2000): 4566

    2000

  30. [30]

    Unbiased expectation values from diffusion quantum Monte Carlo simulations with a fixed number of walkers

    Bos´ a, Ivana, and Stuart M. Rothstein. “Unbiased expectation values from diffusion quantum Monte Carlo simulations with a fixed number of walkers.” TheJournal of Chemical Physics121, no. 10 (2004): 4486-4493

    2004

  31. [31]

    Population size bias in descendant-weighted diffusion quantum Monte Carlo simula- tions

    Warren, G. Lee, and Robert J. Hinde. “Population size bias in descendant-weighted diffusion quantum Monte Carlo simula- tions.”Physical Review E73, no. 5 (2006): 056706

    2006

  32. [32]

    Monte Carlo algorithms for expectation values of coordinate operators

    Barnett, R. N., P. J. Reynolds, and W. A. Lester Jr. “Monte Carlo algorithms for expectation values of coordinate operators.” Journal of Computational Physics96, no. 2 (1991): 258-276

    1991

  33. [33]

    Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers

    Buonaura, Matteo Calandra, and Sandro Sorella. “Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers.”Physical Review B57, no. 18 (1998): 11446

    1998

  34. [34]

    A pure-sampling quantum Monte Carlo algorithm

    Ospadov, Egor, and Stuart M. Rothstein. “A pure-sampling quantum Monte Carlo algorithm.”Journal of Chemical Physics 142, no. 2 (2015)

    2015

  35. [35]

    Reproducibility of fixed-node diffusion Monte Carlo across diverse community codes: The case of water–methane dimer

    Della Pia, Flaviano, Benjamin X. Shi, Yasmine S. Al-Hamdani, Dario Alf´ e, Tyler A. Anderson, Matteo Barborini, Anouar Benali et al. “Reproducibility of fixed-node diffusion Monte Carlo across diverse community codes: The case of water–methane dimer.”Journal of Chemical Physics163, no. 10 (2025)

    2025

  36. [36]

    Variational and diffusion quantum Monte Carlo calculations with the CASINO code

    Needs, R. J., M. D. Towler, N. D. Drummond, P. L´ opez R´ ıos, and J. R. Trail. “Variational and diffusion quantum Monte Carlo calculations with the CASINO code.”Journal of Chemical Physics152, no. 15 (2020)

    2020

  37. [37]

    QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion quantum Monte Carlo

    Kent, Paul RC, Abdulgani Annaberdiyev, Anouar Benali, M. Chandler Bennett, Edgar Josu´ e Landinez Borda, Peter Doak, Hongxia Hao et al. “QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion quantum Monte Carlo.”Journal of Chemical Physics152, no. 17 (2020)

    2020

  38. [38]

    TurboRVB: A many-body toolkit for ab initio electronic simulations by quantum Monte Carlo

    Nakano, Kousuke, Claudio Attaccalite, Matteo Barborini, Luca Capriotti, Michele Casula, Emanuele Coccia, Mario Dagrada et al. “TurboRVB: A many-body toolkit for ab initio electronic simulations by quantum Monte Carlo.”Journal of Chemical Physics152, no. 20 (2020)

    2020

  39. [39]

    PyQMC: An all-Python real-space quantum Monte Carlo module in PySCF

    Wheeler, William A., Shivesh Pathak, Kevin G. Kleiner, Shunyue Yuan, Jo˜ ao NB Rodrigues, Cooper Lorsung, Kittithat Krongchon et al. “PyQMC: An all-Python real-space quantum Monte Carlo module in PySCF.”Journal of Chemical Physics 158, no. 11 (2023)

    2023

  40. [40]

    QMCkl: A kernel library for quantum Monte Carlo applications

    Slootman, Emiel, Vijay Gopal Chilkuri, Aurelien Delval, Max Hoffer, Tommaso Gorni, Fran¸ cois Coppens, Joris van de Nes et al. “QMCkl: A kernel library for quantum Monte Carlo applications.”Journal of Chemical Physics164, no. 11 (2026). Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands Email address:b.j.braams@cwi.nl

    2026