Geometry-Induced Long-Range Correlations in Recurrent Neural Network Quantum States

Asif Bin Ayub; Amine Mohamed Aboussalah; Mohamed Hibat-Allah

quant-phApr 9, 2026

Geometry-Induced Long-Range Correlations in Recurrent Neural Network Quantum States

Authors: Asif Bin Ayub, Amine Mohamed Aboussalah, Mohamed Hibat-Allah

Organizations: Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada · Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada · NYU Tandon School of Engineering, 6 MetroTech Center, Brooklyn, New York 11201, United States · Vector Institute, Toronto, Ontario, M5G 0C6, Canada

Abstract

Neural Quantum States based on autoregressive recurrent neural network (RNN) wave functions enable efficient sampling without Markov-chain autocorrelation, but standard RNN architectures are biased toward finite-length correlations and can fail on states with long-range dependencies. A common response is to adopt transformer-style self-attention, but this typically comes with substantially higher computational and memory overhead. Here we introduce dilated RNN wave functions, where recurrent units access distant sites through dilated connections, injecting an explicit long-range inductive bias while retaining a favorable $\mathcal{O}(N \log N)$ forward pass scaling. We show analytically that dilation changes the correlation geometry and can induce power-law correlation scaling in a simplified linearized and perturbative setting. Numerically, for the critical 1D transverse-field Ising model, dilated RNNs reproduce the expected power-law connected two-point correlations in contrast to the exponential decay typical of conventional RNN ansätze. We further show that the dilated RNN accurately approximates the one-dimensional Cluster state, a paradigmatic example with long-range conditional correlations that has previously been reported to be challenging for RNN-based wave functions. These results highlight dilation as a simple geometric mechanism for building correlation-aware autoregressive neural quantum states.

Figures & tables

Figure 1: Schematic representation of the RNN architectures used. (a) RNN wave function which sequentially generates the conditional probabilities

p_{n}

and conditional phases

\phi_{n}

by taking the one-hot encoded spin

\bm{\sigma}_{n-1}

and hidden vector

\bm{h}_{n-1}

as inputs at the

n^{\text{th}}

step. (b) A dilated RNN illustration for a spin configuration with length

N=5

. For the

n^{\text{th}}

site, the network uses the hidden vector

\bm{h}^{(l)}_{n-2^{l-1}}

from the

(n-2^{l-1})^{\text{th}}

site at the same layer

l

, in addition to

\bm{h}_{n}^{(l-1)}

from the preceding layer. If the dilation length is larger than the spin position

n

, a zero initialization

\bm{h}_{0}

is used instead.

Figure 2: Illustration of the shortest path (in bold green arrows) between the last RNN cell at site $x_{n}$ and the first site $x_{1}$ of (a) the vanilla RNN and (b) the dilated RNN given a system size $N$ . For the vanilla RNN, the shortest path between sites separated by lag $n$ is given by $\mathcal{O}(n)$ , whereas for the dilated RNN, the shortest path has worst-case scaling $\mathcal{O}(\log(n))$ .

Figure 3: (a) A plot of the connected two-point correlation function

C(r)

as a function of distance

r

computed by dilated RNNs for different numbers of layers using the 1D periodic TFIM with

N=100

spins as a testbed. We observe that for

l\geq 4

layers, the expected power-law behavior is captured by our dilated RNN ansatz. Error bars correspond to the one-standard deviation [ 25 ] . (b) A plot of the critical exponent

\eta

versus different numbers of dilation layers obtained from fitting

C(r)

against the chord length

L_{r}

and the corresponding

R^{2}

values.

\eta

approaches the theoretical value of

0.25

for

l\geq 4

layers. The goodness of the fit, indicated by

R^{2}

, is close to unity for those layers. Error bars on

\eta

denote the one-standard-error uncertainty from the covariance matrix of the least-squares fit.

Figure 4: RNN energy expectation values during training on the Cluster state benchmark for a single-layer RNN and a dilated RNN with

6

layers. Note that the dilated architecture converges to the true ground state energy

E_{G}=-64

indicated by the dashed line.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figures	Hyperparameter	Value
Fig. 3	Architecture	1D positive RNN wave function
	Number of samples used for training	$N_{s}=100$
	Number of samples used for estimating correlations	$N_{s}=100,000$
	Number of memory units	$d_{h}=32$
	Training iterations	$100,000$
	Learning rate	$10^{-4}$

Appendix

Table 1: Hyperparameters used to obtain the results reported in this paper.

Neural Quantum States

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Neural Quantum States Variational Monte Carlo Recurrent Neural Networks Quantum Many-Body Physics

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Neural-network quantum states have emerged as a powerful variational framework for quantum many-body systems, with recent progress often driven by massively parallel architectures such as transformers. Recurrent neural network quantum states, however, are frequently regarded as intrinsically sequential and therefore less scalable. Here we revisit this view by showing that modern recurrent architectures can support fast, accurate, and computationally accessible neural quantum state simulations. Using autoregressive recurrent wave functions together with recent advances in parallelizable recurrence, we develop variational ansätze, called parallel scan recurrent neural quantum states (PSR-NQS), which can be trained efficiently within variational Monte Carlo in one and two spatial dimensions. We demonstrate accurate benchmark results and show that, with iterative retraining, our approach reaches two-dimensional spin lattices as large as

52\times52

while remaining in agreement with available quantum Monte Carlo data. Our results establish recurrent architectures as a practical and promising route toward scalable neural quantum state simulations with modest computational resources.

Abstract

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