Pith. sign in

REVIEW 3 major objections 5 minor 4 cited by

Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read By analytically removing PEPS gauge redundancy, the paper makes time-dependent variational Monte Carlo well-conditioned (residuals 1e-9 to 1e-25), enabling accurate 2D real-time evolution through T=12.

desk verdict The gauge-removal idea is right and the Chern benchmark is a genuine proof of concept, but the paper overclaims generality: only one of five examples is exactly checked, the residual diagnostics don't measure representational error, and the abstract promises an Ising test that isn't in the body. read the letter →

arxiv 2512.06768 v5 pith:T2UGBHXU submitted 2025-12-07 cond-mat.str-el cond-mat.stat-mechquant-ph

Yantao Wu , Jannes Nys This is my paper
classification cond-mat.str-elcond-mat.stat-mechquant-ph
keywords projectedentangledpairstatestime-dependentvariationalMonteCarloTDVPgaugeredundancyreal-timedynamicstwo-dimensionalquantumsystemsfractionalHalleffectlatticetheory
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

The paper attempts to show that projected entangled pair states (PEPS), a tensor-network ansatz for two-dimensional quantum systems, can simulate real-time low-energy dynamics for long times if the gauge redundancies of the ansatz are removed analytically before solving the time-dependent variational principle (TDVP) equations. Traditional PEPS dynamical algorithms struggled with ill-conditioned quantum metric tensors, gauge fixing, and loss of positivity; the paper locates the root cause in the many null directions of the log-derivative matrix and eliminates them in closed form by a single QR projection. Once these null directions are removed, the TDVP equation is solved by Cholesky with residuals between 1e-9 and 1e-25, and local quenches on 12x12 and 13x13 lattices run to T=10-12 on one GPU in one to five days. The method is demonstrated on chiral edge propagation, fractional charge pumping (slope 0.5053 vs the expected 1/2), vison confinement in a Z2 lattice gauge theory with Higgs field, superfluid critical velocity, and a global quench of the Ising model at the critical point. If correct, it turns PEPS into a practical classical counterpart to quantum simulators for low-energy 2D dynamics and elementary excitations.

What carries the argument

The key object is the set of PEPS gauge vectors: for each internal virtual bond of the network, an invertible matrix X and its inverse inserted on the two adjacent tensors leaves the wavefunction invariant; the infinitesimal generators form explicit null vectors of the log-derivative matrix O, and hence of the quantum geometric tensor S=O†O. The paper constructs the linear dependence relations among these gauge vectors (generated by lattice plaquettes), adds the global-rescaling and particle-number-conserving null directions, forms a matrix T of all null generators, and performs a complete QR decomposition to obtain a projector Q onto the orthogonal complement. The TDVP equation is then solv

What would settle it

Two checks: (1) insert a known gauge transformation at mid-evolution and verify that physical observables (e.g., site densities) are unchanged within sampling error, confirming the null-space removal; (2) measure the entanglement entropy across a cut of the evolved PEPS and compare with ln(D^2) — if it saturates the bond dimension before T=10, the long-time accuracy claim is uncontrolled.

Watch

Extended reading notes

Core claim

The central discovery is that the historical instability of PEPS time evolution is not intrinsic to the ansatz but to its manifold parametrization: every invertible matrix inserted on a virtual bond leaves the physical state unchanged, generating thousands of null vectors in the quantum geometric tensor that make the TDVP linear system ill-conditioned. These null vectors are exactly describable, so the subspace they span can be removed in closed form via a complete QR decomposition, projecting the TDVP equation onto the complement and leaving a well-conditioned system solvable with a fast Cholesky routine. With regularization only 1e-8, the relative TDVP residual stays between 1e-9 and 1e-25

Load-bearing premise

The simulations assume that the time-evolved state after a local quench remains accurately representable by a PEPS of bond dimension D=4–8 over the full evolution (T=10–12), because the injected energy density is zero and entanglement growth stays area-law-compatible; this is asserted with references and not directly verified for the fractional quantum Hall, lattice gauge theory, or superfluid cases.

Editorial extensions

If this is right

  • Local quench dynamics of 2D lattice models—including fermions, bosons, and gauge theories—can be followed with modest bond dimensions (D=4–8) on 12×12 or 13×13 lattices up to T=10–12 within a few GPU-days.
  • Fractionalized charge transport can be observed directly in real time: the pumped charge per flux cycle from the PEPS–tVMC run is 0.5053, quantitatively confirming the ν=1/2 composite-fermion prediction.
  • Real-time vison dynamics in a Z2 lattice gauge theory correctly distinguishes deconfined (free spreading) from Higgs-phase (bound, confining-string) behavior, demonstrating that gauge-invariant PEPS–tVMC can simulate LGT dynamics classically.
  • A dissipation threshold appears in the superfluid run: at |v|=0.5 total energy is conserved to 2×10^-4 while at |v|=5 energy rises sharply, signaling the Landau critical velocity.
  • Because the TDVP residual is tiny after gauge removal, the practical accuracy limit shifts from the solver to the expressive capacity of the PEPS bond dimension, making further increases in D a direct route to higher fidelity.

Reading between the lines

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

  • Extension: the same gauge-removal logic should transfer to any variational family with exactly known symmetry generators (e.g., translation or particle-number symmetries in neural quantum states), and the small-o memory saving generalizes to any ansatz whose log-derivatives are sample-local, so NQS-tVMC could adopt it for large hidden-unit counts.
  • Extension: the paper's local-quench success suggests PEPS–tVMC may be able to compute two-dimensional spectral functions via time-dependent correlation functions after local perturbations, an observable that static tensor-network methods handle only indirectly.
  • Extension: if the area-law-entanglement assumption is right, one expects a breakdown time t*(D) that grows with bond dimension; measuring how observables converge as D goes from 4 to 8 in the FQHE and gauge-theory runs would provide a scaling test the authors do not report.
  • Extension: the quantized pump slope of 0.5053 was obtained with an explicitly time-dependent flux at finite system size; probing deviations from 1/2 as a function of flux-ramp speed and lattice size would sharpen the claim that this is a true quantized Hall response rather than a near-threshold artifact.
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 / 5 minor

Summary. The paper develops a time-dependent variational Monte Carlo (tVMC) framework for projected entangled pair states (PEPS) on finite lattices. The central technical contribution is the analytic construction of the gauge redundancy of a PEPS: the gauge vectors are shown to be null vectors of the log-derivative matrix, and a single QR projection removes them (along with global-rescaling null vectors) before solving the TDVP equation. This is complemented by a minSR variant and a 'small-o' memory-saving trick. The method is applied to four (or five, depending on counting) local-quench problems on 12×12 or 13×13 lattices: chiral edge propagation in a free-fermion Chern insulator (the only case with an exact benchmark), fractionalized charge pumping in a bosonic Hofstadter model, vison dynamics in a Z2 lattice gauge theory in deconfined and Higgs regimes, and superfluid flow past a moving obstacle. The paper also reports TDVP residuals of 10^-9 to 10^-25, which it interprets as evidence of exceptional numerical stability.

Significance. If the central claim—that analytic gauge removal makes PEPS–tVMC both stable and accurate for low-energy 2D dynamics—is sustained, this would be a valuable methodological advance. The derivation of the gauge null-space structure is parameter-free and the QR-based projection is a clean solution to a known source of ill-conditioning. The paper ships reproducible details (JAX pseudocode, open data) and the Chern-insulator benchmark against exact free-fermion dynamics is genuinely stringent, with density profiles and site-resolved occupations matching to high accuracy. The diversity of model classes is a strength. However, the significance currently rests on more than the conditioning statement: the paper claims accurate dynamics for interacting, topologically ordered, and gauge-theory systems where no exact or converged reference is provided. The evidence for those claims is incomplete, so the significance is real but conditional on additional convergence checks.

major comments (3)
  1. [Sec. IV, Figs. 2–6; SM S-3] Representational error is not controlled for three of the four main examples. Only the Chern insulator (Sec. IV A) is benchmarked against exact free-fermion dynamics. The small residuals in Fig. 6 (Eq. 19) measure how well the Cholesky solve satisfies the sampled TDVP equation; they do not measure the distance between the PEPS manifold and the exact time-evolved state. Energy conservation reported in SM S-3 is expected for TDVP on any manifold when H is time-independent, so it is not a test of bond-dimension sufficiency. Please add D-convergence tests (e.g., D=3,4,5,6) for at least one observable per model, plus sampling error bars, or an exact small-system comparison.
  2. [Sec. IV B, Fig. 3(b)] The FQHE charge-pumping slope of 0.5053 is a topological invariant: any state in the same fractional Hall phase will give a quantized slope under flux insertion, independent of the accuracy of the time-dependent state. Thus this observable is not a sensitive test of PEPS–tVMC. The density profile in Fig. 3(c) is more discriminating, but no D-convergence or error bars are reported; the D'=24 vs 32 check in SM S-2 only concerns total energy. Please show convergence of Q_in(t) and the density profile with D, and include statistical error bars on the slope.
  3. [Secs. IV C, IV D; abstract] The LGT and superfluid results are presented only qualitatively and have no converged reference. The vison propagation in Fig. 4 and the superfluid critical-velocity distinction in Fig. 5 are consistent with physical expectations, but the abstract and Sec. V describe them as demonstrations of accuracy and generality. This is a load-bearing overstatement. Please either add quantitative convergence checks (e.g., observables versus D and dt) or temper the claims to 'qualitative agreement' and 'proof-of-principle'.
minor comments (5)
  1. [Abstract vs. main text] The abstract lists five examples and additionally claims an Ising global-quench simulation, while the main text's Introduction says 'four' examples and no Ising results appear in the body. Align the abstract with the actual content.
  2. [Sec. III C] The sentence 'Note that u1 is not a null direction of O, but only of O' appears to contain a typo; the distinction between O and the centered O is already clear from Eq. (7), so please clarify the wording.
  3. [Fig. 6] The y-axis tick labels are garbled (e.g., '10 26' instead of 10^-26). Ensure the superscripts render correctly.
  4. [SM Table S-2] For the bosonic Hofstadter model, the Hamiltonian is explicitly time-dependent; state in the table that energy conservation is not expected and that D' convergence is used instead. Also clarify whether the wall times are per time step or total, as the notation oscillates.
  5. [Ref. [83]] The data availability link should be accompanied by a persistent identifier or DOI, since GitHub repository links may change.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the gauge-removal construction is self-contained and the benchmarked predictions are not fitted to the target results.

full rationale

The central derivation is not circular. The paper identifies PEPS gauge redundancy analytically: each gauge vector v is constructed to satisfy O v = 0 by the invariance of the PEPS wavefunction under the bond transformation, and the subsequent QR projection simply removes these null directions before solving the TDVP equation. This is a mathematical reduction of the paper's own equations, not an input fitted to the target physics. The exact free-fermion Chern insulator benchmark is an external ground truth, and the fractional Hall pumped charge (0.5053) is compared with the known quantized value, not used to tune the method. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the choice of ansatz. The only self-citations are to the authors' earlier gauge-invariant PEPS sampling schemes (Refs. [31], [52]), which are algorithmic building blocks rather than load-bearing arguments that forbid alternatives; they do not themselves encode the claimed physical results. The residual reported in Eq. (19) is a linear-solve residual, not a fidelity to the exact state, but that is a diagnostic-strength concern, not a circularity. Overall, the paper's derivation chain is self-contained, with at most one minor self-citation that does not affect the logical independence of the central claim.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The ledger contains no invented physical entities. The method assumes standard variational Monte Carlo/TDVP machinery, the previously published gauge-invariant PEPS sampling infrastructure, and the physical assumption that low-energy local quenches keep entanglement within PEPS capacity. Simulation hyperparameters (D, D', N_s, dt, regulator, obstacle details) are chosen by hand and affect the reported accuracy.

free parameters (6)
  • PEPS bond dimension D = D=4 for Chern/FQHE/superfluid; D=8 for Z2 LGT
    Variational truncation chosen by hand to balance cost and accuracy; no systematic convergence study is shown for most examples.
  • Boundary MPS bond dimension D' = D'=16 (Chern, LGT, superfluid), D'=32 (FQHE); D'=24 check for FQHE
    Approximate single-layer contraction accuracy; the effect of D' is only explicitly checked for the bosonic Hofstadter model.
  • Number of Monte Carlo samples N_s = 20480 (40960 for Chern)
    Statistical accuracy of the QGT and gradient estimates; no autocorrelation or sample-convergence analysis is reported.
  • Time step dt = 0.01 (0.02 for superfluid v=5)
    RK4 integrator step; only the superfluid case is checked with two values of dt.
  • Tikhonov regulator epsilon = 1e-8
    Added to the QGT before Cholesky; too large a value biases dynamics, too small risks instability. Chosen by hand.
  • Obstacle potential strength and width (Sec IVD) = strength 2; Gaussian width sqrt(2)
    Chosen to create a localized moving impurity; no variation study, but only qualitative sub/supercritical behavior is claimed.
assumptions (7)
  • domain assumption Time-evolved states after low-energy local quenches preserve area-law entanglement and are representable by PEPS with modest bond dimension
    Invoked throughout Sec IV and explicitly in Sec V; if entanglement grows beyond D=4-8 capacity, the simulations are uncontrolled.
  • domain assumption The null space of the PEPS log-derivative operator O consists exactly of the gauge vectors, the two scaling vectors, with linear dependencies generated by elementary plaquettes
    Sec III asserts this without a formal proof; if additional null directions exist, the QR projection does not fully precondition the TDVP equation.
  • domain assumption Sequential sampling for PEPS satisfies detailed balance and yields unbiased Born-distribution samples
    Sec IV cites Refs. [32,58]; the central statistical estimator relies on this.
  • standard math Standard TDVP/Fubini-Study variational principle with centered QGT gives the optimal local-in-time evolution on the variational manifold
    Sec II equations (1)-(8); well-established background.
  • domain assumption Fermionic PEPS with swap gates correctly encode fermionic statistics
    Sec IV.A cites [58-60] for fermion PEPS; needed for the Chern insulator benchmark.
  • domain assumption The bosonic Hofstadter model at 1/8 filling realizes a bosonic Laughlin-like state with quasiparticle charge 1/2
    Sec IV.B relies on composite-fermion/lattice predictions [62-65,68,69] to interpret the measured slope as a quantized Hall response.
  • domain assumption Z2 lattice gauge theory phase structure and J_c ~ 0.328 from prior literature
    Sec IV.C uses the phase diagram [70,71] to choose deconfined vs Higgs parameters.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo." pith.science (2026). https://pith.science/paper/T2UGBHXU

@misc{pith2026251206768,
  author       = {Pith},
  title        = {Pith review of: Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2UGBHXU}},
  note         = {Machine review of arXiv:2512.06768}
}
read the original abstract

Reliably simulating two-dimensional many-body quantum dynamics with projected entangled pair states (PEPS) has long been a difficult challenge. In this work, we overcome this barrier for low-energy quantum dynamics by developing a stable and efficient time-dependent variational Monte Carlo (tVMC) framework for PEPS. By analytically removing all gauge redundancies of the PEPS manifold and exploiting tensor locality, we obtain a numerically well-conditioned tVMC equation. This enables long-time evolution in previously inaccessible regimes. We explain how the difficulties in the traditional approach, particularly those associated with gauge redundancies, are resolved within tVMC. We demonstrate the power and generality of the method through five representative real-time local quench dynamics in two dimensions: (I) chiral edge propagation in a free-fermion Chern insulator; (II) vison propagation in a pure Z2 gauge theory; (III) vison confinement dynamics in a Z2 lattice gauge theory coupled to Higgs field; (IV) fractionalized charge transport in a fractional Chern insulator; and (V) superfluidity and critical velocity in interacting bosons. All simulations are performed on >= 10 x 10 lattices with evolution times beyond T = 10 using modest computational resources. In addition, we also simulate the paradigmatic dynamics of the Ising model following a global quench at the critical transverse field, and obtain, with modest bond dimension, agreement with previous results. The method significantly extends the reach of classical tensor-network simulations for studying elementary excitations in quantum many-body systems in real-time and provides a valuable computational counterpart to emerging quantum simulators. As a by-product in the development, we also present a new form of minSR, which is more stable and offers a new perspective on tVMC.

Figures

Figures reproduced from arXiv: 2512.06768 by the authors.

Figure 1
Figure 1. FIG. 1: PEPS structure and gauge freedom. (a) A [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Chiral edge dynamics in a 12 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Charge pumping in the bosonic Hofstadter [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Real-time vison dynamics in the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Real-time density response of a 2D [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Residual error [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Efficient classical simulation of two-dimensional long-range systems: Rydberg arrays and beyond

    quant-ph 2026-07 conditional novelty 7.0 of 10

    Adaptive-basis tensor-network VMC reduces local energy evaluation from O(N³) to O(N), enabling simulation of 10×10 dipolar XY Rydberg dynamics.

  2. Resolving support-mismatch by local basis rotation in variational Monte Carlo

    cond-mat.str-el 2026-06 unverdicted novelty 7.0 of 10

    A local basis-rotation sampling scheme resolves support mismatch in variational Monte Carlo for real-time dynamics after local quenches by charged operators.

  3. Spin-charge separation in the triangular-lattice Hofstadter-Hubbard model

    cond-mat.str-el 2026-08 conditional novelty 6.0 of 10

    In the triangular-lattice Hofstadter-Hubbard model at intermediate coupling, injected electrons separate into edge-localized spin and bulk-spreading charge, observed via neural and tensor-network quantum state simulations.

  4. Quantum-classical crossover in fault-tolerant quantum dynamics simulation

    quant-ph 2026-07 conditional novelty 6.0 of 10

    A full-stack projection finds fault-tolerant simulation of mixed-field Ising dynamics overtakes tensor-network and state-vector methods around n = 18-34, with about 2 hours and 3.7 x 10^5 physical qubits at n = 100, p...

Reference graph

Works this paper leans on

86 extracted references · 3 linked inside Pith · cited by 4 Pith papers

  1. [1]

    J. I. Cirac, D. P´ erez-Garc ´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entan- gled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys.93, 045003 (2021)

    2021

  2. [2]

    Xiang,Density Matrix and Tensor Net- work Renormalization(Cambridge University Press, 2023)

    T. Xiang,Density Matrix and Tensor Net- work Renormalization(Cambridge University Press, 2023)

    2023

  3. [3]

    A. W. Sandvik, Computational studies of quantum spin systems, inAIP Conference Proceedings, Vol. 1297 (2010) pp. 135–338

    2010

  4. [5]

    Carleo, F

    G. Carleo, F. Becca, M. Schir´ o, and M. Fabrizio, Lo- calization and glassy dynamics of many-body quan- tum systems, Scientific Reports2, 243 (2012)

    2012

  5. [6]

    Carleo, F

    G. Carleo, F. Becca, L. Sanchez-Palencia, S. Sorella, and M. Fabrizio, Light-cone effect and supersonic correlations in one- and two-dimensional bosonic su- perfluids, Phys. Rev. A89, 031602 (2014)

    2014

  6. [7]

    K. Ido, T. Ohgoe, and M. Imada, Time-dependent many-variable variational monte carlo method for nonequilibrium strongly correlated electron sys- tems, Phys. Rev. B92, 245106 (2015)

    2015

  7. [8]

    Carleo, L

    G. Carleo, L. Cevolani, L. Sanchez-Palencia, and M. Holzmann, Unitary dynamics of strongly inter- acting bose gases with the time-dependent vari- ational monte carlo method in continuous space, Phys. Rev. X7, 031026 (2017)

    2017

  8. [9]

    Sorella, Green function monte carlo with stochas- tic reconfiguration, Phys

    S. Sorella, Green function monte carlo with stochas- tic reconfiguration, Phys. Rev. Lett.80, 4558 (1998)

    1998

Show all 86 references
  1. [10]

    Sorella, Generalized lanczos algorithm for vari- ational quantum monte carlo, Phys

    S. Sorella, Generalized lanczos algorithm for vari- ational quantum monte carlo, Phys. Rev. B64, 024512 (2001)

    2001

  2. [11]

    Becca and S

    F. Becca and S. Sorella,Quantum Monte Carlo Ap- proaches for Correlated Systems(Cambridge Uni- versity Press, 2017)

    2017

  3. [12]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science355, 602 (2017)

    2017

  4. [13]

    Schmitt and M

    M. Schmitt and M. Heyl, Quantum many-body dy- namics in two dimensions with artificial neural net- works, Phys. Rev. Lett.125, 100503 (2020)

    2020

  5. [14]

    I. L. Guti´ errez and C. B. Mendl, Real time evolution with neural-network quantum states, Quantum6, 627 (2022)

    2022

  6. [15]

    Medvidovi´ c and D

    M. Medvidovi´ c and D. Sels, Variational quantum dynamics of two-dimensional rotor models, PRX Quantum4, 040302 (2023)

    2023

  7. [16]

    J. Nys, G. Pescia, A. Sinibaldi, and G. Carleo, Ab-initio variational wave functions for the time- dependent many-electron schr¨ odinger equation, Na- ture Communications15, 9404 (2024)

    2024

  8. [17]

    Sharir, A

    O. Sharir, A. Shashua, and G. Carleo, Neural tensor contractions and the expressive power of deep neu- ral quantum states, Physical Review B106, 205136 (2022)

    2022

  9. [18]

    G. Hu, W. Ouyang, C.-Y. Lu, C. Lin, and H.-S. Zhong, Efficient and universal neural-network de- coder for stabilizer-based quantum error correction, arXiv preprint arXiv:2502.19971 (2025)

    2025 arXiv

  10. [19]

    Sinibaldi, A

    A. Sinibaldi, A. F. Mello, M. Collura, and G. Carleo, Non-stabilizerness of neural quantum states, arXiv preprint arXiv:2502.09725 (2025)

    2025

  11. [20]

    Sinibaldi, C

    A. Sinibaldi, C. Giuliani, G. Carleo, and F. Vicen- tini, Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution, Quantum7, 1131 (2023)

    2023

  12. [21]

    Gravina, V

    L. Gravina, V. Savona, and F. Vicentini, Neural projected quantum dynamics: a systematic study, Quantum9, 1803 (2025)

    2025

  13. [22]

    Lange, A

    H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, From architectures to applications: a review of neural quantum states, Quantum Science and Technology9, 040501 (2024)

    2024

  14. [23]

    Verstraete and J

    F. Verstraete and J. I. Cirac, Renormalization al- gorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066 (2004)

    2004 arXiv

  15. [24]

    Jordan, R

    J. Jordan, R. Or´ us, G. Vidal, F. Verstraete, and J. I. Cirac, Classical simulation of infinite-size quantum lattice systems in two spatial dimensions, Phys. Rev. Lett.101, 250602 (2008)

    2008

  16. [26]

    Lubasch, J

    M. Lubasch, J. I. Cirac, and M.-C. Ba˜ nuls, Algo- rithms for finite projected entangled pair states, Phys. Rev. B , 064425 (2014)

    2014

  17. [27]

    Liu, S.-S

    W.-Y. Liu, S.-S. Gong, Y.-B. Li, D. Poilblanc, W.- Q. Chen, and Z.-C. Gu, Gapless quantum spin liquid and global phase diagram of the spin-1/2J 1 −J 2 square antiferromagnetic Heisenberg model, Science Bulletin67, 1034 (2022)

    2022

  18. [28]

    W.-Y. Liu, J. Hasik, S.-S. Gong, D. Poilblanc, W.- Q. Chen, and Z.-C. Gu, Emergence of gapless quan- 11 tum spin liquid from deconfined quantum critical point, Phys. Rev. X12, 031039 (2022)

    2022

  19. [29]

    Liu, X.-T

    W.-Y. Liu, X.-T. Zhang, Z. Wang, S.-S. Gong, W.-Q. Chen, and Z.-C. Gu, Quantum criticality with emergent symmetry in the extended shastry- sutherland model, Phys. Rev. Lett.133, 026502 (2024)

    2024

  20. [30]

    W.-Y. Liu, H. Zhai, R. Peng, Z.-C. Gu, and G. K.- L. Chan, Accurate simulation of the hubbard model with finite fermionic projected entangled pair states, Phys. Rev. Lett.134, 256502 (2025)

    2025

  21. [31]

    Wu and W.-Y

    Y. Wu and W.-Y. Liu, Accurate gauge-invariant tensor-network simulations for abelian lattice gauge theory in (2 + 1)D: Ground-state and real-time dy- namics, Phys. Rev. Lett.135, 130401 (2025)

    2025

  22. [32]

    Liu, Y.-Z

    W.-Y. Liu, Y.-Z. Huang, S.-S. Gong, and Z.-C. Gu, Accurate simulation for finite projected entangled pair states in two dimensions, Phys. Rev. B103, 235155 (2021)

    2021

  23. [33]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys.82, 277 (2010)

    2010

  24. [34]

    Calabrese and J

    P. Calabrese and J. Cardy, Quantum quenches in 1+1 dimensional conformal field theories, Journal of Statistical Mechanics: Theory and Experiment 2016, 064003 (2016)

    2016

  25. [35]

    Wu, Dissipative dynamics in isolated quan- tum spin chains after a local quench (2020), arXiv:2010.00700 [cond-mat.stat-mech]

    Y. Wu, Dissipative dynamics in isolated quan- tum spin chains after a local quench (2020), arXiv:2010.00700 [cond-mat.stat-mech]

    2020 arXiv

  26. [36]

    Fischer, M

    O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Scanning tunneling spectroscopy of high-temperature superconductors, Rev. Mod. Phys.79, 353 (2007)

    2007

  27. [37]

    J. A. Sobota, Y. He, and Z.-X. Shen, Angle-resolved photoemission spectroscopy, Rev. Mod. Phys.93, 025006 (2021)

    2021

  28. [38]

    Haegeman, C

    J. Haegeman, C. Lubich, I. Oseledets, B. Vander- eycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B94, 165116 (2016)

    2016

  29. [39]

    Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys

    G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003)

    2003

  30. [40]

    M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore, and F. Pollmann, Time-evolving a matrix product state with long-ranged interactions, Phys. Rev. B91, 165112 (2015)

    2015

  31. [41]

    Vidal, Classical simulation of infinite-size quan- tum lattice systems in one spatial dimension, Phys

    G. Vidal, Classical simulation of infinite-size quan- tum lattice systems in one spatial dimension, Phys. Rev. Lett.98, 070201 (2007)

    2007

  32. [42]

    Czarnik and J

    P. Czarnik and J. Dziarmaga, Time evolution of an infinite projected entangled pair state: An algo- rithm from first principles, Phys. Rev. B98, 045110 (2018)

    2018

  33. [43]

    Czarnik, J

    P. Czarnik, J. Dziarmaga, and P. Corboz, Time evo- lution of an infinite projected entangled pair state: An efficient algorithm, Phys. Rev. B99, 035115 (2019)

    2019

  34. [44]

    Hubig and J

    C. Hubig and J. I. Cirac, Time-dependent study of disordered models with infinite projected entangled pair states, SciPost Phys.6, 031 (2019)

    2019

  35. [45]

    Hubig, A

    C. Hubig, A. Bohrdt, M. Knap, F. Grusdt, and J. I. Cirac, Evaluation of time-dependent correlators af- ter a local quench in iPEPS: hole motion in the t-J model, SciPost Phys.8, 021 (2020)

    2020

  36. [46]

    S.-H. Lin, M. P. Zaletel, and F. Pollmann, Efficient simulation of dynamics in two-dimensional quantum spin systems with isometric tensor networks, Phys. Rev. B106, 245102 (2022)

    2022

  37. [47]

    Dziarmaga, Time evolution of an infinite pro- jected entangled pair state: A gradient tensor up- date in the tangent space, Phys

    J. Dziarmaga, Time evolution of an infinite pro- jected entangled pair state: A gradient tensor up- date in the tangent space, Phys. Rev. B106, 014304 (2022)

    2022

  38. [48]

    R. T. Ponnaganti, M. Mambrini, and D. Poilblanc, Real-time dynamics of a critical resonating valence bond spin liquid, Phys. Rev. B106, 195132 (2022)

    2022

  39. [49]

    Tindall and M

    J. Tindall and M. Fishman, Gauging tensor net- works with belief propagation, SciPost Phys.15, 222 (2023)

    2023

  40. [50]

    J. D. Arias Espinoza and P. Corboz, Spectral func- tions with infinite projected entangled-pair states, Phys. Rev. B110, 094314 (2024)

    2024

  41. [51]

    Motoyama, T

    Y. Motoyama, T. Okubo, K. Yoshimi, S. Morita, T. Aoyama, T. Kato, and N. Kawashima, Tenes-v2: Enhancement for real-time and finite temperature simulations of quantum many-body systems, Com- puter Physics Communications315, 109692 (2025)

    2025

  42. [52]

    Z. Dai, Y. Wu, T. Wang, and M. P. Zaletel, Fermionic isometric tensor network states in two di- mensions, Phys. Rev. Lett.134, 026502 (2025)

    2025

  43. [53]

    J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics76, 289 (1980)

    1980

  44. [54]

    J. C. Butcher,Numerical Methods for Ordinary Dif- ferential Equations, 3rd ed. (Wiley, 2016)

    2016

  45. [55]

    Chen and M

    A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Na- ture Physics20, 1476 (2024)

    2024

  46. [56]

    Wu, Supplemental material (2025)

    Y. Wu, Supplemental material (2025)

    2025

  47. [57]

    M. P. Zaletel and F. Pollmann, Isometric tensor net- work states in two dimensions, Phys. Rev. Lett.124, 037201 (2020)

    2020

  48. [58]

    Wu and Z

    Y. Wu and Z. Dai, Algorithms for variational monte carlo calculations of fermion projected entangled pair states in the swap gates formulation and the detailed balance of tensor network sequential sam- pling (2025), arXiv:2506.20106 [cond-mat.str-el]

    2025

  49. [59]

    C. V. Kraus, N. Schuch, F. Verstraete, and J. I. Cirac, Fermionic projected entangled pair states, Phys. Rev. A81, 052338 (2010)

    2010

  50. [60]

    Corboz, R

    P. Corboz, R. Or´ us, B. Bauer, and G. Vidal, Simu- lation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states, Phys. Rev. B81, 165104 (2010)

    2010

  51. [61]

    D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two- dimensional magnetotransport in the extreme quan- tum limit, Phys. Rev. Lett.48, 1559 (1982)

    1982

  52. [62]

    R. B. Laughlin, Anomalous quantum hall effect: An incompressible quantum fluid with fraction- ally charged excitations, Phys. Rev. Lett.50, 1395 (1983)

    1983

  53. [63]

    J. K. Jain, Composite-fermion approach for the frac- 12 tional quantum hall effect, Phys. Rev. Lett.63, 199 (1989)

    1989

  54. [64]

    A. S. Sørensen, E. Demler, and M. D. Lukin, Frac- tional quantum hall states of atoms in optical lat- tices, Phys. Rev. Lett.94, 086803 (2005)

    2005

  55. [65]

    M¨ oller and N

    G. M¨ oller and N. R. Cooper, Composite fermion theory for bosonic quantum hall states on lattices, Phys. Rev. Lett.103, 105303 (2009)

    2009

  56. [66]

    Regnault and B

    N. Regnault and B. A. Bernevig, Fractional chern insulator, Phys. Rev. X1, 021014 (2011)

    2011

  57. [67]

    We thank Bartholomew Andrews for bringing this practice to our attention

  58. [68]

    R. B. Laughlin, Quantized hall conductivity in two dimensions, Phys. Rev. B23, 5632 (1981)

    1981

  59. [69]

    D. J. Thouless, Quantization of particle transport, Phys. Rev. B27, 6083 (1983)

    1983

  60. [70]

    Fradkin and S

    E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with higgs fields, Phys. Rev. D19, 3682 (1979)

    1979

  61. [71]

    F. Wu, Y. Deng, and N. Prokof’ev, Phase diagram of the toric code model in a parallel magnetic field, Phys. Rev. B85, 195104 (2012)

    2012

  62. [72]

    E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, P. Hauke, M. Dal- monte, T. Monz, P. Zoller, and R. Blatt, Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature534, 516 (2016)

    2016

  63. [73]

    B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Observation of gauge invariance in a 71-site Bose-Hubbard quantum simulator, Nature587, 392 (2020)

    2020

  64. [74]

    M. Meth, J. Zhang, J. F. Haase, C. Edmunds, L. Postler, A. J. Jena, A. Steiner, L. Dellanto- nio, R. Blatt, P. Zoller, T. Monz, P. Schindler, C. Muschik, and M. Ringbauer, Simulating two- dimensional lattice gauge theories on a qudit quan- tum computer, Nature Physics21, 570 (2025)

    2025

  65. [75]

    A. W. Sandvik, Stochastic series expansion method for quantum spin systems, Phys. Rev. B59, R14157 (1999)

    1999

  66. [76]

    Heinrichs and W

    S. Heinrichs and W. J. Mullin, Quantum-monte- carlo calculations for bosons in a two-dimensional harmonic trap, Journal of Low Temperature Physics 113, 231 (1998)

    1998

  67. [77]

    Landau, Theory of the superfluidity of helium ii, Phys

    L. Landau, Theory of the superfluidity of helium ii, Phys. Rev.60, 356 (1941)

    1941

  68. [78]

    G. H. Golub and C. F. Van Loan,Matrix Computa- tions - 4th Edition(Johns Hopkins University Press, Philadelphia, PA, 2013)

    2013

  69. [79]

    Zhang, H

    M. Zhang, H. Zhang, C. Wang, and L. He, Scal- able tensor network algorithm for thermal quantum many-body systems in two dimensions, Phys. Rev. B111, 075146 (2025)

    2025

  70. [80]

    Hauschild and F

    J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor network python (tenpy), SciPost Physics Lecture Notes , 005 (2018)

    2018

  71. [81]

    Hauschild, J

    J. Hauschild, J. Unfried, S. Anand, B. Andrews, M. Bintz, U. Borla, S. Divic, M. Drescher, J. Geiger, M. Hefel, K. H´ emery, W. Kadow, J. Kemp, N. Kirchner, V. S. Liu, G. M¨ oller, D. Parker, M. Rader, A. Romen, S. Scalet, L. Schoonderwoerd, M. Schulz, T. Soejima, P. Thoma, Y....

    2024

  72. [82]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018)

    2018

  73. [83]

    Wu, github data repository,https: //github.com/yantaow/open_data/tree/main/ wu2025real-time(2025)

    Y. Wu, github data repository,https: //github.com/yantaow/open_data/tree/main/ wu2025real-time(2025)

    2025

  74. [84]

    Con- siderOmatrix with shape (N s, Np)

    The small-otrick is the following process. Con- siderOmatrix with shape (N s, Np). When comput- ing the minSR matrixOO †, to save memory, one can partition the parameters intoKparts of the same size: i) compute log-derivatives of 1/Kof the paramters, and form a smaller matrixo...

  75. [85]

    One thus only keeps o[x](s)lrdu =O[x](s) s(x)lrdu This saves memory by a factor ofd

    For tensor network states (TNS), for given samplesand at a given lattice sitex,O[x](s) plrdu = 0 if p̸=s(x). One thus only keeps o[x](s)lrdu =O[x](s) s(x)lrdu This saves memory by a factor ofd. We indexowitho sa, wherearuns over a selected subset of the parameter sets, dependi...

  76. [86]

    P α OsαO† αs′: To compute this fromo, one constructs a sample tensorps.t.p sa =s(x) wheresis the sam- ple indexed bysandxis the site associated with the parameter indexed bya. Then X α OsαO† αs′ = X a osao† as′(psa ==p s′a) = K−1X k=0 X a osao† as′(psa ==k)&(p s′a ==k) = K−1X ...

  77. [87]

    P α Osα ⟨O⟩∗ α = 1 Ns P α Osα P s′ O∗ s′α = 1 Ns P s′(OO†)ss′ =jnp.average(OOdag, axis=1)

  78. [88]

    numpy as jnp 2def g e t _ m i n S R _ m a t r i x ( OOdag ) : 3r 4Input : 5OOdag : ( Ns , Ns ) , obtained from g et _O Od ag () 6 7OOavg_c = jnp

    Overall, the minSR matrix is obtained as Listing 2: JAX implementation of computing the full minSR matrix 1import jax . numpy as jnp 2def g e t _ m i n S R _ m a t r i x ( OOdag ) : 3r 4Input : 5OOdag : ( Ns , Ns ) , obtained from g et _O Od ag () 6 7OOavg_c = jnp . average ( ...

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.