arxiv: 2604.05032 · v1 · submitted 2026-04-06 · 🪐 quant-ph · cond-mat.other

Recognition: 2 theorem links

· Lean Theorem

Real-time Dynamics in 3D for up to 1000 Qubits with Neural Quantum States: Quenches and the Quantum Kibble--Zurek Mechanism

Vighnesh Dattatraya Naik , Zheng-Hang Sun , Markus Heyl

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:44 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.other

keywords neural quantum statesquantum Kibble-Zurek mechanism3D Ising modelreal-time dynamicsquench dynamicsvariational methodscritical phenomenamany-body entanglement

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The pith

Neural quantum states simulate 3D dynamics up to 1000 qubits and confirm the quantum Kibble-Zurek mechanism

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

The paper aims to establish that a residual-based convolutional neural quantum state architecture can accurately represent the time evolution of three-dimensional quantum systems with up to 1000 qubits, overcoming the rapid entanglement growth that defeats standard methods beyond one dimension. By focusing on the transverse-field Ising model, the authors simulate various quench protocols, including sudden quenches to the quantum critical point. They derive logarithmic corrections to the scaling laws from two-loop renormalization-group equations and demonstrate robust data collapse for the correlation function, excess energy, and quantum Fisher information across all simulated sizes. This yields the first large-scale numerical verification of the 3D quantum Kibble-Zurek mechanism at the upper critical dimension. The result positions these states as a scalable tool for nonequilibrium studies in three-dimensional quantum matter.

Core claim

The residual-based convolutional neural quantum state provides a variational representation of the many-body wave function that reliably captures real-time dynamics in the 3D transverse-field Ising model on lattices up to 1000 qubits. For finite-rate quenches to the critical point, incorporating logarithmic corrections from renormalization-group flow equations produces data collapse for the correlation function, excess energy, and quantum Fisher information, with agreement to the expected scaling dimensions of the 3D quantum Kibble-Zurek mechanism.

What carries the argument

Residual-based convolutional neural quantum state architecture on cubic spin lattices, which variationally approximates the time-evolved wave function to enable dynamics simulations where entanglement grows rapidly.

If this is right

NQS can now be applied to explore other nonequilibrium phenomena in 3D quantum systems at large scales.
The quantum Fisher information accesses universal multipartite entanglement dynamics during critical quenches.
Logarithmic corrections must be accounted for when analyzing scaling at the upper critical dimension.
Distinct regimes such as collapse-and-revival dynamics are accessible with the same framework.

Where Pith is reading between the lines

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

The convolutional architecture may generalize to other 3D lattice models or geometries for broader critical dynamics studies.
These large-scale results could provide benchmarks for experimental 3D quantum simulators once such devices reach comparable sizes.
Further refinements might permit longer evolution times or larger lattices to investigate additional nonequilibrium effects.

Load-bearing premise

The neural quantum state maintains controlled variational accuracy for the time-evolved wave function during quenches to the critical point despite rapid entanglement growth.

What would settle it

Absence of data collapse across system sizes when the observables are rescaled with the derived forms that include logarithmic corrections, or mismatch between simulated values and the predicted scaling dimensions, would show that the representation fails to capture the dynamics.

Figures

Figures reproduced from arXiv: 2604.05032 by Markus Heyl, Vighnesh Dattatraya Naik, Zheng-Hang Sun.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Correlation-function collapse for different system [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Universal dynamics of multipartite entanglement. (a) Time evolution of the rescaled QFI density [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Convergence of NQS for collapse-and-revival dynamics. The panels illustrate the numerical convergence of the NQS [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Convergence of NQS for dynamics following a sudden quench to criticality. The panels illustrate the numerical [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Kibble–Zurek scaling with a linear ramp protocol for [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

read the original abstract

Exponential complexity of many-body wave functions limits accurate numerical simulations of real-time dynamics, especially beyond 1D, where rapid entanglement growth poses severe challenges. Neural Quantum States (NQS) have emerged as a powerful approach for real-time dynamics in 2D, but their scalability and accuracy in 3D have remained an open challenge. Here, we establish NQS as a scalable framework for 3D quantum dynamics by introducing a residual-based convolutional architecture tailored to cubic spin lattices. Focusing on the 3D transverse-field Ising model, we demonstrate that NQS reliably capture distinct quench regimes, including collapse-and-revival dynamics and, most challengingly, the dynamics following a sudden quench to the quantum critical point. We perform finite-rate quenches to the critical point on lattices containing up to $1000$ qubits, an unprecedented system size for numerical simulations of real-time dynamics beyond 1D. This enables the first large-scale numerical demonstration of the 3D quantum Kibble--Zurek mechanism. The QKZM in 3D is particularly intriguing because it lies at the upper critical dimension of the Ising universality class, where the standard power laws are modified by logarithmic factors together with prominent sub-leading logarithmic corrections. By deriving these corrections from renormalization-group flow equations up to two-loop order, we obtain a robust data collapse across all simulated system sizes for the correlation function, the excess energy, and the quantum Fisher information, the latter revealing universal multipartite-entanglement dynamics. In all cases, we find compelling agreement with the expected scaling dimensions. Our findings establish NQS as a scalable and reliable tool for exploring nonequilibrium phenomena in 3D quantum matter and for providing numerical benchmarks for 3D quantum simulators.

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

NQS reaches 1000 qubits for 3D Ising quenches and shows QKZM data collapse with two-loop log corrections, but variational accuracy lacks tight quantitative checks.

read the letter

The main thing to know is that this paper gets a residual convolutional neural quantum state to run real-time dynamics on the 3D transverse-field Ising model up to 1000 sites, including sudden quenches to criticality, and extracts scaling for the correlation function, excess energy, and quantum Fisher information that collapses with the logarithmic corrections expected at the upper critical dimension. They derive those corrections from standard two-loop RG flow equations, independent of the simulations, then show the numerics line up with the predicted dimensions across system sizes. That combination is new: earlier NQS dynamics work stayed mostly in 2D or smaller lattices, and this is the first explicit large-scale 3D QKZM test with those corrections included. The architecture choice for cubic lattices and the coverage of different quench regimes, from collapse-revival to critical, are handled cleanly. The results look internally consistent and give concrete benchmarks that could matter for 3D quantum simulators. The soft spot is the control on variational error. Critical quenches drive fast entanglement growth, and without reported infidelity bounds, energy variances, or direct comparisons to exact methods on smaller systems that scale up, it is not obvious the NQS representation stays faithful enough at the largest sizes and times to guarantee the collapse is not partly an artifact. The abstract claims robust agreement, but if the full text does not add those diagnostics, the central claim rests on an assumption that needs more evidence. This is for people working on numerical methods for quantum many-body dynamics or on nonequilibrium critical phenomena beyond one dimension. A reader who wants scalable NQS examples or 3D Kibble-Zurek tests would get value from the scaling plots and architecture details. It deserves a serious referee because the system sizes and the independent RG comparison are substantial steps forward, even if revisions will likely focus on error analysis and convergence. I would send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a residual convolutional Neural Quantum State (NQS) architecture for real-time dynamics of the 3D transverse-field Ising model. It simulates quenches on lattices up to 1000 qubits, captures collapse-and-revival and critical-quench regimes, and provides the first large-scale numerical demonstration of the 3D quantum Kibble-Zurek mechanism. Logarithmic corrections to scaling (derived from two-loop RG flow equations) are tested via data collapse of the correlation function, excess energy, and quantum Fisher information, yielding agreement with expected scaling dimensions.

Significance. If the variational accuracy is confirmed, the work establishes NQS as a scalable tool for 3D nonequilibrium quantum dynamics at sizes far beyond prior reach. It supplies the first numerical test of QKZM at the upper critical dimension of the Ising class, where logarithmic corrections dominate, and generates concrete benchmarks for quantum simulators. The RG-derived corrections and cross-quantity data collapse constitute a strong, falsifiable prediction that the numerics can validate.

major comments (1)

[Results section on QKZM data collapse] The central claim that the residual convolutional NQS yields a faithful representation of the time-evolved state up to 1000 qubits (enabling the reported data collapse) is load-bearing but lacks quantitative support. No energy-variance bounds, infidelity estimates, or convergence diagnostics are supplied for the largest lattices, nor are direct comparisons to exact results shown for intermediate sizes to demonstrate that variational bias remains controlled during critical quenches where entanglement grows rapidly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment regarding quantitative validation of the NQS accuracy below, and commit to revisions that strengthen this aspect without altering the core claims.

read point-by-point responses

Referee: [Results section on QKZM data collapse] The central claim that the residual convolutional NQS yields a faithful representation of the time-evolved state up to 1000 qubits (enabling the reported data collapse) is load-bearing but lacks quantitative support. No energy-variance bounds, infidelity estimates, or convergence diagnostics are supplied for the largest lattices, nor are direct comparisons to exact results shown for intermediate sizes to demonstrate that variational bias remains controlled during critical quenches where entanglement grows rapidly.

Authors: We agree that the original manuscript would benefit from explicit diagnostics to support the variational fidelity claim. While exact infidelity is inaccessible for lattices beyond small sizes due to exponential complexity, we monitored the energy variance throughout training and time evolution for all system sizes up to 1000 qubits; these variances converge to low, stable values consistent with the observed dynamics. For intermediate sizes (up to 4x4x4 = 64 sites), we have additional benchmarks against exact diagonalization and matrix-product-state simulations that confirm the NQS reproduces collapse-and-revival features and critical-quench observables with controlled deviations. The multi-observable data collapse itself across system sizes serves as a strong consistency check against uncontrolled bias. In the revised manuscript we will add (i) energy-variance time series for representative large lattices, (ii) direct small-system comparisons, and (iii) a brief discussion of training convergence criteria. revision: yes

Circularity Check

0 steps flagged

No significant circularity; RG-derived scalings tested by independent NQS numerics

full rationale

The paper derives the two-loop logarithmic corrections to the 3D Ising QKZM scaling from standard renormalization-group flow equations, independent of any simulation data. These predictions are then confronted with NQS-generated observables (correlation function, excess energy, QFI) via data collapse across system sizes up to 1000 qubits. No step reduces a claimed prediction to a fitted parameter, self-definition, or load-bearing self-citation; the NQS architecture and variational accuracy are presented as numerical tools whose fidelity is assessed by agreement with the external RG theory rather than by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the variational accuracy of the NQS ansatz for time-evolved states and the applicability of standard Ising-model renormalization-group equations; no new free parameters or invented entities are introduced beyond the neural architecture.

axioms (1)

standard math Renormalization-group flow equations for the Ising universality class up to two-loop order
Invoked to derive the logarithmic corrections that modify the Kibble-Zurek scaling at the upper critical dimension.

pith-pipeline@v0.9.0 · 5642 in / 1421 out tokens · 49019 ms · 2026-05-10T19:44:06.923284+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/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

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

We establish NQS as a scalable framework for 3D quantum dynamics by introducing a residual-based convolutional architecture tailored to cubic spin lattices... first large-scale numerical demonstration of the 3D quantum Kibble–Zurek mechanism... at the upper critical dimension d+z=4
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

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

By deriving these corrections from renormalization-group flow equations up to two-loop order, we obtain a robust data collapse... for the correlation function, the excess energy, and the quantum Fisher information

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

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Collapse-and-revival dynamics To assess the numerical reliability of the NQS approach in the collapse-and-revival regime, we study the conver- gence of the dynamics with increasing network depthn. Figure 6 shows results for a 5×5×5 lattice for three ob- servables: the transverse magnetization⟨σ x i (t)⟩, the lon- gitudinal magnetization⟨σ z i (t)⟩, and th...
[2]

Figure 7 compares results on a 6×6×6 lattice obtained with network depthsn= 3 andn= 4

A sudden quench to the critical point We next examine the convergence of the NQS dynam- ics following a sudden quench to the quantum critical point. Figure 7 compares results on a 6×6×6 lattice obtained with network depthsn= 3 andn= 4. We analyze the time evolution of the transverse magnetiza- tion⟨σ x i (t)⟩, the correlation function⟨σ z 0σz d⟩ − ⟨σ z 0⟩...
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Finite-rate quenches – QKZM For the finite-rate quenches discussed in the main text, we consider system sizes fromL= 5 toL= 10. We use network depthn= 3 for most simulations, except forL= 5, where we usen= 2. The time evolution is performed fromt=−1.5τ q tot= 0, witht= 0 mark- ing the point at which the ramp reaches criticality. We consider quench times i...
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