Variational subspace methods and application to improving variational Monte Carlo dynamics

Adrien Kahn; Filippo Vicentini; Luca Gravina

arxiv: 2507.08930 · v2 · submitted 2025-07-11 · 🪐 quant-ph · cond-mat.dis-nn

Variational subspace methods and application to improving variational Monte Carlo dynamics

Adrien Kahn , Luca Gravina , Filippo Vicentini This is my paper

Pith reviewed 2026-05-19 04:43 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nn

keywords variational subspace methodsvariational Monte Carloquantum dynamicsdiscretization errorssubspace optimizationdeterminant state mappingexcited states estimationpost-processing

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The pith

A formalism for direct subspace optimization enables Bridge to mitigate discretization errors in variational Monte Carlo dynamics.

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

The paper introduces a formalism that permits the direct manipulation and optimization of subspaces in variational methods rather than optimizing states individually. By employing the determinant state mapping, it extends concepts like distance, energy, and Monte Carlo sampling to entire subspaces, which also recovers a known method for estimating excited states. As an application, it proposes the Bridge technique that forms linear combinations of variational time-evolved states to correct for errors introduced by time discretization in dynamics simulations. This approach is computationally cheap and can be applied as a post-processing step, making it valuable for improving simulations of quantum systems where exact dynamics are hard to compute.

Core claim

The central claim is that a determinant state mapping extends variational notions of distance and energy to subspaces, enabling direct optimization over linear combinations of states and the recovery of excited-state methods, with the Bridge procedure then using such combinations of time-evolved variational states to reduce errors from time discretization in Monte Carlo dynamics.

What carries the argument

The determinant state mapping that extends distance, energy, and Monte Carlo estimators from states to subspaces, together with the Bridge procedure that extracts optimal linear combinations from discretized variational trajectories.

Load-bearing premise

The linear combinations extracted from variational time-evolved states remain faithful approximations to the true dynamics and do not introduce uncontrolled biases when used to correct discretization error.

What would settle it

Compare Bridge-corrected trajectories against exact time evolution on a small solvable system such as a few-qubit transverse-field Ising chain and verify whether observable errors decrease faster with effective time-step refinement than in the uncorrected variational case.

Figures

Figures reproduced from arXiv: 2507.08930 by Adrien Kahn, Filippo Vicentini, Luca Gravina.

**Figure 2.** Figure 2: Comparison between the final infidelity er [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: Infidelity to exact dynamics and magnetization on the [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Performance of Bridge using families of p [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Magnetization on the 8 × 8 transverse-field Ising model with h = 0.1hc (a) and h = 2hc (b). The original states are vision transformers optimized with p-tVMC using the scheme SLPE2 with time step Jδ = 0.05 (a) and hδ = 0.025 (b). The reference for (a) are vision transformers optimized with p-tVMC using the higher order scheme SLPE3 with a smaller time step Jδ = 0.025. The reference for (b) is convolutional… view at source ↗

**Figure 6.** Figure 6: (a) Comparison between the final infidelity error of Bridge performed using various estimators of the [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗

read the original abstract

We present a formalism that allows for the direct manipulation and optimization of subspaces, circumventing the need to optimize individual states when using subspace methods. Using the determinant state mapping, we can naturally extend notions such as distance and energy to subspaces, as well as Monte Carlo estimators, recovering the excited states estimation method proposed by Pfau et al. As a practical application, we then introduce Bridge, a method that improves the performance of variational dynamics by extracting linear combinations of variational time-evolved states. We find that Bridge is both computationally inexpensive and capable of significantly mitigating the errors that arise from discretizing the dynamics, and can thus be systematically used as a post-processing tool for variational dynamics.

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.

Desk Editor's Note private letter to a colleague

The paper gives a subspace formalism that recovers an existing estimator and adds a cheap post-processing step called Bridge to reduce discretization errors in variational Monte Carlo dynamics.

read the letter

The main thing to know is that this work supplies a direct formalism for handling subspaces in variational Monte Carlo and then uses it to build Bridge, a post-processing correction that combines linear combinations of time-evolved states to cut discretization artifacts in dynamics runs. The authors map states to determinants so they can define distances, energies, and Monte Carlo estimators over whole subspaces instead of single states. This recovers the excited-state estimator from Pfau et al. but extends it to the dynamics setting as the new practical contribution. Bridge itself is presented as inexpensive and effective at mitigating the errors that come from finite time steps, which is a common pain point in these simulations. That part is straightforward and could be adopted without major changes to existing codebases. The formalism looks clean on its own terms and avoids obvious circularity by building on the determinant mapping and standard variational machinery. The soft spot is the lack of a clear argument or test showing that the extracted combinations reduce the leading discretization bias rather than trading it for a new bias tied to the subspace projection or the variational ansatz itself. If the time-step errors correlate with the choice of subspace, the correction could amplify rather than suppress the discrepancy with continuous evolution. The paper would benefit from either an a-priori bound or concrete comparisons against exact dynamics on small systems to settle this. This is aimed at researchers already running variational Monte Carlo for time-dependent problems in quantum many-body physics or chemistry. A reader working on dynamics or error control in these methods would find the construction useful to try. It deserves a serious referee because the idea is practical and the underlying formalism is reproducible in principle, even if the error-mitigation claim needs tighter validation.

Referee Report

2 major / 2 minor

Summary. The paper presents a formalism for the direct manipulation and optimization of subspaces using the determinant state mapping. This extends notions of distance, energy, and Monte Carlo estimators to subspaces and recovers the excited-state estimation method of Pfau et al. As a practical application, the authors introduce the Bridge method, which extracts linear combinations of variational time-evolved states to improve variational Monte Carlo dynamics by mitigating discretization errors. They claim Bridge is computationally inexpensive and can be used systematically as a post-processing tool for variational dynamics.

Significance. If the central claims hold, the subspace formalism could offer a useful generalization for handling multiple states in variational methods without separate optimizations, with potential applications in excited-state calculations and quantum dynamics. The Bridge method addresses a practical limitation in time-discretized variational simulations, which are common in quantum many-body physics. Its value as a post-processing tool would be notable if the error mitigation is shown to be systematic rather than incidental, but the overall significance remains moderate pending rigorous validation of bias control.

major comments (2)

[Bridge method] Bridge method (application section): The central claim that linear combinations of variational time-evolved states systematically mitigate discretization errors lacks an a priori error bound or analysis demonstrating that the subspace projection reduces (rather than trades or amplifies) the leading discretization bias. This is load-bearing because the method relies on the combinations remaining faithful to the underlying continuous dynamics; without such a bound or a counter-example showing faster convergence than the raw variational trajectory, the post-processing utility is not established.
[Formalism] Monte Carlo estimators for subspaces (formalism section): It is unclear whether the unbiased Monte Carlo estimators for the combination coefficients remain reliable when the input variational states already carry systematic bias from both the ansatz and the discrete time-stepping. If the subspace projection correlates with the discretization error (e.g., through shared parameters), the extracted coefficients could introduce uncontrolled errors, undermining the claim of systematic improvement.

minor comments (2)

[Abstract] The abstract states that Bridge 'significantly mitigating the errors' but does not reference specific quantitative benchmarks, figures, or tables that would allow immediate assessment of the improvement magnitude.
[Formalism] Notation for subspace distance and energy could be clarified with explicit definitions or comparisons to the single-state case to improve readability for readers familiar with standard VMC.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us clarify the presentation of both the subspace formalism and the Bridge method. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [Bridge method] Bridge method (application section): The central claim that linear combinations of variational time-evolved states systematically mitigate discretization errors lacks an a priori error bound or analysis demonstrating that the subspace projection reduces (rather than trades or amplifies) the leading discretization bias. This is load-bearing because the method relies on the combinations remaining faithful to the underlying continuous dynamics; without such a bound or a counter-example showing faster convergence than the raw variational trajectory, the post-processing utility is not established.

Authors: We agree that an a priori error bound would provide stronger theoretical support. The manuscript presents the Bridge method as a post-processing technique that finds the optimal linear combination within the subspace spanned by the variational time-evolved states, and we demonstrate through numerical results on several model systems that this combination reduces the observed discretization error relative to the raw trajectories. In the revised manuscript we have added a discussion of the conditions under which the projection is expected to reduce rather than amplify bias, based on the variational states remaining reasonable approximations to the continuous dynamics. We do not claim a general proof of systematic improvement for arbitrary ansatzes and time steps, but the empirical evidence supports its practical utility. revision: partial
Referee: [Formalism] Monte Carlo estimators for subspaces (formalism section): It is unclear whether the unbiased Monte Carlo estimators for the combination coefficients remain reliable when the input variational states already carry systematic bias from both the ansatz and the discrete time-stepping. If the subspace projection correlates with the discretization error (e.g., through shared parameters), the extracted coefficients could introduce uncontrolled errors, undermining the claim of systematic improvement.

Authors: The Monte Carlo estimators derived in the formalism section are unbiased for the subspace quantities (overlaps, energies, etc.) evaluated on the supplied set of variational states, irrespective of how those states were generated. Any bias present in the individual states—whether from the ansatz or from time discretization—is inherited by the subspace; the Bridge procedure simply optimizes the combination coefficients within that subspace. We have revised the text to emphasize that the method is a post-processing step applied to already-generated trajectories and to include a brief analysis showing that, in the cases examined, the extracted coefficients do not amplify the leading error. If strong correlations between discretization errors and the subspace exist, the improvement may be limited, but this does not invalidate the unbiased character of the estimators themselves. revision: yes

standing simulated objections not resolved

A rigorous a priori error bound establishing that the subspace projection systematically reduces (rather than merely trades) the leading discretization bias for general variational dynamics.

Circularity Check

0 steps flagged

No significant circularity detected; derivation builds on external mappings and prior methods

full rationale

The paper defines a subspace formalism via the determinant state mapping to extend distance, energy, and Monte Carlo estimators, then recovers the Pfau et al. excited-state method before introducing Bridge as a post-processing linear combination step for variational dynamics. No step reduces a claimed prediction or improvement to a fitted parameter or definition drawn from the same data by construction; the central claims rest on independent optimization within the subspace rather than self-referential re-labeling of inputs. The derivation chain remains self-contained against the cited external machinery without load-bearing self-citations or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the determinant state mapping and standard variational Monte Carlo assumptions; no new free parameters or invented physical entities are introduced in the abstract.

axioms (2)

domain assumption Determinant state mapping preserves the necessary algebraic structure for defining subspace distances and energies
Invoked when the authors state they can naturally extend notions such as distance and energy to subspaces via the determinant state mapping.
domain assumption Variational time-evolved states remain sufficiently accurate to allow useful linear combinations for error correction
Underlying the claim that Bridge can systematically mitigate discretization errors.

pith-pipeline@v0.9.0 · 5640 in / 1433 out tokens · 43114 ms · 2026-05-19T04:43:33.432939+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose to encode the subspace V = span({|ϕk⟩}k) with the quantum state |ϕA⟩ given by the antisymmetrization of a given basis, |ϕA⟩ = S− |ϕ1⟩ ⊗ … ⊗ |ϕm⟩ … the mapping is known in mathematics as the Plücker embedding.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The eigenvalues µk of G−1G(H) … satisfy a generalized variational principle … Ek ≤ µk … we call G−1G(H) the Rayleigh matrix.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Bridge … extracts linear combinations of variational time-evolved states … α(t) = e−i G−1G(H) t α(0).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

[1]

Qma-complete prob- lems

Adam D. Bookatz. “Qma-complete prob- lems”. Quantum Information and Compu- tation 14, 361–383 (2014)

work page 2014
[2]

Hand-waving and interpretive dance: an introductory course on tensor net- works

Jacob C Bridgeman and Christopher T Chubb. “Hand-waving and interpretive dance: an introductory course on tensor net- works”. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017)

work page 2017
[3]

A practical introduction to tensor networks: Matrix product states and projected entangled pair states

Rom´ an Or´ us. “A practical introduction to tensor networks: Matrix product states and projected entangled pair states”. Annals of Physics 349, 117–158 (2014)

work page 2014
[4]

The density-matrix renormalization group in the age of matrix product states

Ulrich Schollw¨ ock. “The density-matrix renormalization group in the age of matrix product states”. Annals of Physics 326, 96–192 (2011)

work page 2011
[5]

The variational quantum eigen- solver: A review of methods and best prac- tices

Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Ed- ward Grant, Leonard Wossnig, Ivan Rung- ger, George H. Booth, and Jonathan Ten- nyson. “The variational quantum eigen- solver: A review of methods and best prac- tices”. Physics Reports 986, 1–128 (2022)

work page 2022
[6]

A variational eigen- value solver on a photonic quantum proces- sor

Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Al´ an Aspuru-Guzik, and Jeremy L. O’Brien. “A variational eigen- value solver on a photonic quantum proces- sor”. Nature Communications 5 (2014)

work page 2014
[7]

The theory of variational hybrid quantum- classical algorithms

Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Al´ an Aspuru-Guzik. “The theory of variational hybrid quantum- classical algorithms”. New Journal of Physics 18, 023023 (2016)

work page 2016
[8]

Solving the quantum many-body problem with artificial neural networks

Giuseppe Carleo and Matthias Troyer. “Solving the quantum many-body problem with artificial neural networks”. Science 355, 602–606 (2017)

work page 2017
[9]

Modern ap- plications of machine learning in quantum sciences

Anna Dawid, Julian Arnold, Borja Re- quena, Alexander Gresch, Marcin P lodzie´ n, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam B¨ uttner, Robert Oku la, Gorka Mu˜ noz-Gil, Ro- drigo A. Vargas-Hern´ andez, Alba Cervera- Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabri´ e, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicen...

work page 2022
[10]

An area law for one- dimensional quantum systems

M B Hastings. “An area law for one- dimensional quantum systems”. Journal of Statistical Mechanics: Theory and Experi- ment 2007, P08024–P08024 (2007)

work page 2007
[11]

Two-dimensional frustrated j1-j2 model studied with neural network quantum states

Kenny Choo, Titus Neupert, and Giuseppe Carleo. “Two-dimensional frustrated j1-j2 model studied with neural network quantum states”. Physical Review B 100 (2019)

work page 2019
[12]

Transformer varia- tional wave functions for frustrated quan- tum spin systems

Luciano Loris Viteritti, Riccardo Rende, and Federico Becca. “Transformer varia- tional wave functions for frustrated quan- tum spin systems”. Physical Review Let- ters130 (2023)

work page 2023
[13]

Empower- ing deep neural quantum states through ef- ficient optimization

Ao Chen and Markus Heyl. “Empower- ing deep neural quantum states through ef- ficient optimization”. Nature Physics 20, 1476–1481 (2024)

work page 2024
[14]

Accu- rate neural quantum states for interacting lattice bosons

Zakari Denis and Giuseppe Carleo. “Accu- rate neural quantum states for interacting lattice bosons” (2024)

work page 2024
[15]

Back- flow transformations via neural networks for quantum many-body wave functions

Di Luo and Bryan K. Clark. “Back- flow transformations via neural networks for quantum many-body wave functions”. Phys- ical Review Letters122 (2019)

work page 2019
[16]

Fermionic wave functions from neural- network constrained hidden states

Javier Robledo Moreno, Giuseppe Car- leo, Antoine Georges, and James Stokes. “Fermionic wave functions from neural- network constrained hidden states”. Pro- ceedings of the National Academy of Sci- ences119 (2022)

work page 2022
[17]

Unitary dynamics of strongly interacting bose gases with the time-dependent vari- ational monte-carlo method in continuous space

Giuseppe Carleo, Lorenzo Cevolani, Laurent Sanchez-Palencia, and Markus Holzmann. “Unitary dynamics of strongly interacting bose gases with the time-dependent vari- ational monte-carlo method in continuous space”. Physical Review X 7 (2017)

work page 2017
[18]

Quan- tum many-body dynamics in two dimensions with artificial neural networks

Markus Schmitt and Markus Heyl. “Quan- tum many-body dynamics in two dimensions with artificial neural networks”. Physical Review Letters125 (2020)

work page 2020
[19]

Highly resolved spectral functions of two-dimensional systems with neural quantum states

Tiago Mendes-Santos, Markus Schmitt, and Markus Heyl. “Highly resolved spectral functions of two-dimensional systems with neural quantum states”. Phys. Rev. Lett. 131, 046501 (2023)

work page 2023
[20]

Unbiasing time-dependent variational monte carlo by projected quantum evolu- tion

Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini. “Unbiasing time-dependent variational monte carlo by projected quantum evolu- tion”. Quantum 7, 1131 (2023)

work page 2023
[21]

Neural projected quantum dynamics: a systematic study

Luca Gravina, Vincenzo Savona, and Fil- ippo Vicentini. “Neural projected quantum dynamics: a systematic study” (2024)

work page 2024
[22]

Many-body dynamics with explicitly time-dependent neural quan- tum states

Anka Van de Walle, Markus Schmitt, and Annabelle Bohrdt. “Many-body dynamics with explicitly time-dependent neural quan- tum states” (2024)

work page 2024
[23]

Time-dependent neural galerkin method for quantum dynamics

Alessandro Sinibaldi, Douglas Hendry, Fil- ippo Vicentini, and Giuseppe Carleo. “Time-dependent neural galerkin method for quantum dynamics” (2024)

work page 2024
[24]

Ma- trix computations

G.H. Golub and C.F. Van Loan. “Ma- trix computations”. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press. (2013). url: https://books.google.fr/books?id= X5YfsuCWpxMC

work page 2013
[25]

Numerical methods for large eigenvalue problems: Revised edition

Yousef Saad. “Numerical methods for large eigenvalue problems: Revised edition”. Soci- ety for Industrial and Applied Mathematics. (2011)

work page 2011
[26]

Numerical lin- ear algebra

L.N. Trefethen and D. Bau. “Numerical lin- ear algebra”. Other Titles in Applied Math- ematics. Society for Industrial and Applied Mathematics. (1997). url: https://books. google.fr/books?id=4Mou5YpRD_kC

work page 1997
[27]

Arnoldi-lindblad time evolution: Faster- than-the-clock algorithm for the spectrum of time-independent and floquet open quantum systems

Fabrizio Minganti and Dolf Huybrechts. “Arnoldi-lindblad time evolution: Faster- than-the-clock algorithm for the spectrum of time-independent and floquet open quantum systems”. Quantum 6, 649 (2022)

work page 2022
[28]

Generalized lanczos method for systematic optimization of neural-network quantum states

Jia-Qi Wang, Rong-Qiang He, and Zhong- Yi Lu. “Generalized lanczos method for systematic optimization of neural-network quantum states” (2025)

work page 2025
[29]

System- atic improvement of neural network quan- tum states using a lanczos recursion

Hongwei Chen, Douglas Hendry, Phillip Weinberg, and Adrian E. Feiguin. “System- atic improvement of neural network quan- tum states using a lanczos recursion” (2022)

work page 2022
[30]

Generalized lanczos algo- rithm for variational quantum monte carlo

Sandro Sorella. “Generalized lanczos algo- rithm for variational quantum monte carlo”. Physical Review B64 (2001)

work page 2001
[31]

Time-evolution methods for matrix- product states

Sebastian Paeckel, Thomas K¨ ohler, An- dreas Swoboda, Salvatore R. Manmana, 16 Ulrich Schollw¨ ock, and Claudius Hu- big. “Time-evolution methods for matrix- product states”. Annals of Physics 411, 167998 (2019)

work page 2019
[32]

Variational perturbation theory in open quantum systems for efficient steady state computation

Andr´ e Melo, Gaspard Beugnot, and Fabrizio Minganti. “Variational perturbation theory in open quantum systems for efficient steady state computation” (2025)

work page 2025
[33]

Symmetries and many-body excitations with neural-network quantum states

Kenny Choo, Giuseppe Carleo, Nicolas Reg- nault, and Titus Neupert. “Symmetries and many-body excitations with neural-network quantum states”. Physical Review Let- ters121 (2018)

work page 2018
[34]

Accurate computation of quan- tum excited states with neural networks

David Pfau, Simon Axelrod, Halvard Sut- terud, Ingrid von Glehn, and James S. Spencer. “Accurate computation of quan- tum excited states with neural networks”. Science385 (2024)

work page 2024
[35]

Princi- ples of algebraic geometry

Phillip Griffiths and Joseph Harris. “Princi- ples of algebraic geometry”. Wiley. (1994)

work page 1994
[36]

Amplitude Ratios and Neural Network Quantum States

Vojtech Havlicek. “Amplitude Ratios and Neural Network Quantum States”. Quan- tum 7, 938 (2023)

work page 2023
[37]

Ground and excited states from ensemble variational principles

Lexin Ding, Cheng-Lin Hong, and Christian Schilling. “Ground and excited states from ensemble variational principles”. Quantum 8, 1525 (2024)

work page 2024
[38]

A fast and efficient algorithm for slater de- terminant updates in quantum monte carlo simulations

Phani K. V. V. Nukala and P. R. C. Kent. “A fast and efficient algorithm for slater de- terminant updates in quantum monte carlo simulations”. The Journal of Chemical Physics130 (2009)

work page 2009
[39]

Colloquium: Eigenvector con- tinuation and projection-based emulators

Thomas Duguet, Andreas Ekstr¨ om, Richard J. Furnstahl, Sebastian K¨ onig, and Dean Lee. “Colloquium: Eigenvector con- tinuation and projection-based emulators”. Reviews of Modern Physics 96 (2024)

work page 2024
[40]

Inter- polating numerically exact many-body wave functions for accelerated molecular dynam- ics

Yannic Rath and George H. Booth. “Inter- polating numerically exact many-body wave functions for accelerated molecular dynam- ics”. Nature Communications 16 (2025)

work page 2025
[41]

Netket: A machine learning toolkit for many-body quantum systems

Giuseppe Carleo, Kenny Choo, Damian Hof- mann, James E.T. Smith, Tom Wester- hout, Fabien Alet, Emily J. Davis, Stavros Efthymiou, Ivan Glasser, Sheng-Hsuan Lin, Marta Mauri, Guglielmo Mazzola, Chris- tian B. Mendl, Evert van Nieuwenburg, Os- sian O’Reilly, Hugo Th´ eveniaut, Giacomo Torlai, Filippo Vicentini, and Alexander Wi- etek. “Netket: A machine le...

work page 2019
[42]

Netket 3: Machine learning toolbox for many-body quantum systems

Filippo Vicentini, Damian Hofmann, At- tila Szab´ o, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calder´ on, Nikita As- trakhantsev, and Giuseppe Carleo. “Netket 3: Machine learning toolbox for many-body quantum systems”. SciPost Physics Code- bases (2022)

work page 2022
[43]

mpi4jax: Zero-copy mpi communica- tion of jax arrays

Dion H¨ afner and Filippo Vicentini. “mpi4jax: Zero-copy mpi communica- tion of jax arrays”. Journal of Open Source Software 6, 3419 (2021)

work page 2021
[44]

JAX: composable transformations of Python+NumPy programs

James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. “JAX: composable transformations of Python+NumPy programs” (2018)

work page 2018
[45]

Equinox: neural networks in jax via callable pytrees and filtered transforma- tions

Patrick Kidger and Cristian Garcia. “Equinox: neural networks in jax via callable pytrees and filtered transforma- tions” (2021)

work page 2021
[46]

Flax: A neural network library and ecosys- tem for JAX

Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Ronde- pierre, Andreas Steiner, and Marc van Zee. “Flax: A neural network library and ecosys- tem for JAX” (2024)

work page 2024
[47]

Qutip: An open-source python framework for the dynamics of open quan- tum systems

J.R. Johansson, P.D. Nation, and Franco Nori. “Qutip: An open-source python framework for the dynamics of open quan- tum systems”. Computer Physics Commu- nications 183, 1760–1772 (2012)

work page 2012
[48]

Qutip 2: A python framework for the dynamics of open quantum sys- tems

J.R. Johansson, P.D. Nation, and Franco Nori. “Qutip 2: A python framework for the dynamics of open quantum sys- tems”. Computer Physics Communications 184, 1234–1240 (2013)

work page 2013
[49]

mp- math: a Python library for arbitrary- precision floating-point arithmetic (version 1.3.0)

The mpmath development team. “mp- math: a Python library for arbitrary- precision floating-point arithmetic (version 1.3.0)”. https://mpmath.org/ (2023). 17 A Determinant state formalism A.1 Antisymmetric subspace and antisymmetrizer We recall the definitions of the antisymmetric subspace and the antisymmetrizer operator. Definition A.1 (Permutation opera...

work page 2023

[1] [1]

Qma-complete prob- lems

Adam D. Bookatz. “Qma-complete prob- lems”. Quantum Information and Compu- tation 14, 361–383 (2014)

work page 2014

[2] [2]

Hand-waving and interpretive dance: an introductory course on tensor net- works

Jacob C Bridgeman and Christopher T Chubb. “Hand-waving and interpretive dance: an introductory course on tensor net- works”. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017)

work page 2017

[3] [3]

A practical introduction to tensor networks: Matrix product states and projected entangled pair states

Rom´ an Or´ us. “A practical introduction to tensor networks: Matrix product states and projected entangled pair states”. Annals of Physics 349, 117–158 (2014)

work page 2014

[4] [4]

The density-matrix renormalization group in the age of matrix product states

Ulrich Schollw¨ ock. “The density-matrix renormalization group in the age of matrix product states”. Annals of Physics 326, 96–192 (2011)

work page 2011

[5] [5]

The variational quantum eigen- solver: A review of methods and best prac- tices

Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Ed- ward Grant, Leonard Wossnig, Ivan Rung- ger, George H. Booth, and Jonathan Ten- nyson. “The variational quantum eigen- solver: A review of methods and best prac- tices”. Physics Reports 986, 1–128 (2022)

work page 2022

[6] [6]

A variational eigen- value solver on a photonic quantum proces- sor

Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Al´ an Aspuru-Guzik, and Jeremy L. O’Brien. “A variational eigen- value solver on a photonic quantum proces- sor”. Nature Communications 5 (2014)

work page 2014

[7] [7]

The theory of variational hybrid quantum- classical algorithms

Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Al´ an Aspuru-Guzik. “The theory of variational hybrid quantum- classical algorithms”. New Journal of Physics 18, 023023 (2016)

work page 2016

[8] [8]

Solving the quantum many-body problem with artificial neural networks

Giuseppe Carleo and Matthias Troyer. “Solving the quantum many-body problem with artificial neural networks”. Science 355, 602–606 (2017)

work page 2017

[9] [9]

Modern ap- plications of machine learning in quantum sciences

Anna Dawid, Julian Arnold, Borja Re- quena, Alexander Gresch, Marcin P lodzie´ n, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam B¨ uttner, Robert Oku la, Gorka Mu˜ noz-Gil, Ro- drigo A. Vargas-Hern´ andez, Alba Cervera- Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabri´ e, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicen...

work page 2022

[10] [10]

An area law for one- dimensional quantum systems

M B Hastings. “An area law for one- dimensional quantum systems”. Journal of Statistical Mechanics: Theory and Experi- ment 2007, P08024–P08024 (2007)

work page 2007

[11] [11]

Two-dimensional frustrated j1-j2 model studied with neural network quantum states

Kenny Choo, Titus Neupert, and Giuseppe Carleo. “Two-dimensional frustrated j1-j2 model studied with neural network quantum states”. Physical Review B 100 (2019)

work page 2019

[12] [12]

Transformer varia- tional wave functions for frustrated quan- tum spin systems

Luciano Loris Viteritti, Riccardo Rende, and Federico Becca. “Transformer varia- tional wave functions for frustrated quan- tum spin systems”. Physical Review Let- ters130 (2023)

work page 2023

[13] [13]

Empower- ing deep neural quantum states through ef- ficient optimization

Ao Chen and Markus Heyl. “Empower- ing deep neural quantum states through ef- ficient optimization”. Nature Physics 20, 1476–1481 (2024)

work page 2024

[14] [14]

Accu- rate neural quantum states for interacting lattice bosons

Zakari Denis and Giuseppe Carleo. “Accu- rate neural quantum states for interacting lattice bosons” (2024)

work page 2024

[15] [15]

Back- flow transformations via neural networks for quantum many-body wave functions

Di Luo and Bryan K. Clark. “Back- flow transformations via neural networks for quantum many-body wave functions”. Phys- ical Review Letters122 (2019)

work page 2019

[16] [16]

Fermionic wave functions from neural- network constrained hidden states

Javier Robledo Moreno, Giuseppe Car- leo, Antoine Georges, and James Stokes. “Fermionic wave functions from neural- network constrained hidden states”. Pro- ceedings of the National Academy of Sci- ences119 (2022)

work page 2022

[17] [17]

Unitary dynamics of strongly interacting bose gases with the time-dependent vari- ational monte-carlo method in continuous space

Giuseppe Carleo, Lorenzo Cevolani, Laurent Sanchez-Palencia, and Markus Holzmann. “Unitary dynamics of strongly interacting bose gases with the time-dependent vari- ational monte-carlo method in continuous space”. Physical Review X 7 (2017)

work page 2017

[18] [18]

Quan- tum many-body dynamics in two dimensions with artificial neural networks

Markus Schmitt and Markus Heyl. “Quan- tum many-body dynamics in two dimensions with artificial neural networks”. Physical Review Letters125 (2020)

work page 2020

[19] [19]

Highly resolved spectral functions of two-dimensional systems with neural quantum states

Tiago Mendes-Santos, Markus Schmitt, and Markus Heyl. “Highly resolved spectral functions of two-dimensional systems with neural quantum states”. Phys. Rev. Lett. 131, 046501 (2023)

work page 2023

[20] [20]

Unbiasing time-dependent variational monte carlo by projected quantum evolu- tion

Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini. “Unbiasing time-dependent variational monte carlo by projected quantum evolu- tion”. Quantum 7, 1131 (2023)

work page 2023

[21] [21]

Neural projected quantum dynamics: a systematic study

Luca Gravina, Vincenzo Savona, and Fil- ippo Vicentini. “Neural projected quantum dynamics: a systematic study” (2024)

work page 2024

[22] [22]

Many-body dynamics with explicitly time-dependent neural quan- tum states

Anka Van de Walle, Markus Schmitt, and Annabelle Bohrdt. “Many-body dynamics with explicitly time-dependent neural quan- tum states” (2024)

work page 2024

[23] [23]

Time-dependent neural galerkin method for quantum dynamics

Alessandro Sinibaldi, Douglas Hendry, Fil- ippo Vicentini, and Giuseppe Carleo. “Time-dependent neural galerkin method for quantum dynamics” (2024)

work page 2024

[24] [24]

Ma- trix computations

G.H. Golub and C.F. Van Loan. “Ma- trix computations”. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press. (2013). url: https://books.google.fr/books?id= X5YfsuCWpxMC

work page 2013

[25] [25]

Numerical methods for large eigenvalue problems: Revised edition

Yousef Saad. “Numerical methods for large eigenvalue problems: Revised edition”. Soci- ety for Industrial and Applied Mathematics. (2011)

work page 2011

[26] [26]

Numerical lin- ear algebra

L.N. Trefethen and D. Bau. “Numerical lin- ear algebra”. Other Titles in Applied Math- ematics. Society for Industrial and Applied Mathematics. (1997). url: https://books. google.fr/books?id=4Mou5YpRD_kC

work page 1997

[27] [27]

Arnoldi-lindblad time evolution: Faster- than-the-clock algorithm for the spectrum of time-independent and floquet open quantum systems

Fabrizio Minganti and Dolf Huybrechts. “Arnoldi-lindblad time evolution: Faster- than-the-clock algorithm for the spectrum of time-independent and floquet open quantum systems”. Quantum 6, 649 (2022)

work page 2022

[28] [28]

Generalized lanczos method for systematic optimization of neural-network quantum states

Jia-Qi Wang, Rong-Qiang He, and Zhong- Yi Lu. “Generalized lanczos method for systematic optimization of neural-network quantum states” (2025)

work page 2025

[29] [29]

System- atic improvement of neural network quan- tum states using a lanczos recursion

Hongwei Chen, Douglas Hendry, Phillip Weinberg, and Adrian E. Feiguin. “System- atic improvement of neural network quan- tum states using a lanczos recursion” (2022)

work page 2022

[30] [30]

Generalized lanczos algo- rithm for variational quantum monte carlo

Sandro Sorella. “Generalized lanczos algo- rithm for variational quantum monte carlo”. Physical Review B64 (2001)

work page 2001

[31] [31]

Time-evolution methods for matrix- product states

Sebastian Paeckel, Thomas K¨ ohler, An- dreas Swoboda, Salvatore R. Manmana, 16 Ulrich Schollw¨ ock, and Claudius Hu- big. “Time-evolution methods for matrix- product states”. Annals of Physics 411, 167998 (2019)

work page 2019

[32] [32]

Variational perturbation theory in open quantum systems for efficient steady state computation

Andr´ e Melo, Gaspard Beugnot, and Fabrizio Minganti. “Variational perturbation theory in open quantum systems for efficient steady state computation” (2025)

work page 2025

[33] [33]

Symmetries and many-body excitations with neural-network quantum states

Kenny Choo, Giuseppe Carleo, Nicolas Reg- nault, and Titus Neupert. “Symmetries and many-body excitations with neural-network quantum states”. Physical Review Let- ters121 (2018)

work page 2018

[34] [34]

Accurate computation of quan- tum excited states with neural networks

David Pfau, Simon Axelrod, Halvard Sut- terud, Ingrid von Glehn, and James S. Spencer. “Accurate computation of quan- tum excited states with neural networks”. Science385 (2024)

work page 2024

[35] [35]

Princi- ples of algebraic geometry

Phillip Griffiths and Joseph Harris. “Princi- ples of algebraic geometry”. Wiley. (1994)

work page 1994

[36] [36]

Amplitude Ratios and Neural Network Quantum States

Vojtech Havlicek. “Amplitude Ratios and Neural Network Quantum States”. Quan- tum 7, 938 (2023)

work page 2023

[37] [37]

Ground and excited states from ensemble variational principles

Lexin Ding, Cheng-Lin Hong, and Christian Schilling. “Ground and excited states from ensemble variational principles”. Quantum 8, 1525 (2024)

work page 2024

[38] [38]

A fast and efficient algorithm for slater de- terminant updates in quantum monte carlo simulations

Phani K. V. V. Nukala and P. R. C. Kent. “A fast and efficient algorithm for slater de- terminant updates in quantum monte carlo simulations”. The Journal of Chemical Physics130 (2009)

work page 2009

[39] [39]

Colloquium: Eigenvector con- tinuation and projection-based emulators

Thomas Duguet, Andreas Ekstr¨ om, Richard J. Furnstahl, Sebastian K¨ onig, and Dean Lee. “Colloquium: Eigenvector con- tinuation and projection-based emulators”. Reviews of Modern Physics 96 (2024)

work page 2024

[40] [40]

Inter- polating numerically exact many-body wave functions for accelerated molecular dynam- ics

Yannic Rath and George H. Booth. “Inter- polating numerically exact many-body wave functions for accelerated molecular dynam- ics”. Nature Communications 16 (2025)

work page 2025

[41] [41]

Netket: A machine learning toolkit for many-body quantum systems

Giuseppe Carleo, Kenny Choo, Damian Hof- mann, James E.T. Smith, Tom Wester- hout, Fabien Alet, Emily J. Davis, Stavros Efthymiou, Ivan Glasser, Sheng-Hsuan Lin, Marta Mauri, Guglielmo Mazzola, Chris- tian B. Mendl, Evert van Nieuwenburg, Os- sian O’Reilly, Hugo Th´ eveniaut, Giacomo Torlai, Filippo Vicentini, and Alexander Wi- etek. “Netket: A machine le...

work page 2019

[42] [42]

Netket 3: Machine learning toolbox for many-body quantum systems

Filippo Vicentini, Damian Hofmann, At- tila Szab´ o, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calder´ on, Nikita As- trakhantsev, and Giuseppe Carleo. “Netket 3: Machine learning toolbox for many-body quantum systems”. SciPost Physics Code- bases (2022)

work page 2022

[43] [43]

mpi4jax: Zero-copy mpi communica- tion of jax arrays

Dion H¨ afner and Filippo Vicentini. “mpi4jax: Zero-copy mpi communica- tion of jax arrays”. Journal of Open Source Software 6, 3419 (2021)

work page 2021

[44] [44]

JAX: composable transformations of Python+NumPy programs

James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. “JAX: composable transformations of Python+NumPy programs” (2018)

work page 2018

[45] [45]

Equinox: neural networks in jax via callable pytrees and filtered transforma- tions

Patrick Kidger and Cristian Garcia. “Equinox: neural networks in jax via callable pytrees and filtered transforma- tions” (2021)

work page 2021

[46] [46]

Flax: A neural network library and ecosys- tem for JAX

Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Ronde- pierre, Andreas Steiner, and Marc van Zee. “Flax: A neural network library and ecosys- tem for JAX” (2024)

work page 2024

[47] [47]

Qutip: An open-source python framework for the dynamics of open quan- tum systems

J.R. Johansson, P.D. Nation, and Franco Nori. “Qutip: An open-source python framework for the dynamics of open quan- tum systems”. Computer Physics Commu- nications 183, 1760–1772 (2012)

work page 2012

[48] [48]

Qutip 2: A python framework for the dynamics of open quantum sys- tems

J.R. Johansson, P.D. Nation, and Franco Nori. “Qutip 2: A python framework for the dynamics of open quantum sys- tems”. Computer Physics Communications 184, 1234–1240 (2013)

work page 2013

[49] [49]

mp- math: a Python library for arbitrary- precision floating-point arithmetic (version 1.3.0)

The mpmath development team. “mp- math: a Python library for arbitrary- precision floating-point arithmetic (version 1.3.0)”. https://mpmath.org/ (2023). 17 A Determinant state formalism A.1 Antisymmetric subspace and antisymmetrizer We recall the definitions of the antisymmetric subspace and the antisymmetrizer operator. Definition A.1 (Permutation opera...

work page 2023