Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.24337

New non-Euclidean neural quantum states from additional types of hyperbolic recurrent neural networks

H. L. Dao

quant-phcond-mat.dis-nncs.LGSubmitted Apr 27, 2026• Announced Apr 28, 2026• 0 citations• Revised Jul 16, 2026 (v2)
v1 · Apr 27, 2026v2 · Jul 16, 2026v3 · Aug 3, 2026
arXivHTMLPDF

Abstract

In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincare hyperbolic GRU, to new variants including Poincare RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing the new non-Euclidean hyperbolic NQS ansatzes, we generalize the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincare GRU NQS when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the Heisenberg $J_1J_2$ and $J_1J_2J_3$ models. Here, using larger systems consisting of 100 spins, we find that all four hyperbolic RNN/GRU NQS variants always outperform their respective Euclidean counterpart with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different $J_2$ and ($J_2, J_3$) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatzes are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Poincare RNN/GRU and Lorentz RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.

Comments
v2: additional improved results added, new discussions added, main conclusions remain unchanged

Related papers

Two-dimensional Hyperbolic RNN Neural Quantum State

H. L. Dao

In the first part of this work, we construct the first type of two-dimensional (2D) hyperbolic neural quantum state (NQS) in the form of the Lorentz 2DRNN (Recurrent Neural Network) and benchmark its performance against the Euclidean 2DRNN in the paradigmatic $N \times N$ 2D Transverse Field Ising Model (2DTFIM) setting with different lattice sizes up to N = 12 at different transverse magnetic field strengths. We find that hyperbolic Lorentz 2DRNN NQS robustly outperforms Euclidean 2DRNN NQS in our experiments when the system is at the phase transition point, where the physics can be described by a conformal field theory (CFT) - known via the AdS/CFT holographic duality to possess an underlying hyperbolic geometry. In the second part of this work, we benchmark the performances of recently introduced one-dimensional (1D) hyperbolic NQS ansatzes (including Poincare RNN/GRU and Lorentz RNN/GRU) against their Euclidean versions in $N \times N$ 2DTFIM, converted to a one-dimensional sequence. The findings extend our previous results showing that 1D hyperbolic NQS robustly outperforms 1D Euclidean NQS, thanks to the combined effects of the induced 1st and Nth neighbor interaction hierarchy and the CFT physics at criticality. While studies on larger system sizes remain future work, this study serves as a proof-of-concept for the utility, effectiveness, and superior performance of 1D and 2D hyperbolic NQS ansatzes compared to conventional Euclidean NQS models in many-body quantum systems - particularly those exhibiting structural hierarchy, criticality, or a combination of both.

quant-pharXiv:2606.25600•Jun 25, 2026•0 citations

Hyperbolic Restricted Boltzmann Machine Neural Quantum State

H. L. Dao

We construct the first type of non-Euclidean non-autoregressive neural quantum state (NQS) in the form of the hyperbolic Restricted Boltzmann Machine (HRBM), which is studied in the variational Monte-Carlo (VMC) setting of the Quantum Sherrington-Kirkpatrick (QSK) model whose ground state exhibits volume-law entanglement. Across a 512-fold increase in the Hilbert space dimension corresponding to a system size increase from $N=14$ to $N=24$, HRBM NQS robustly outperforms its Euclidean version, the RBM NQS, in terms of better ground state energy optimization as well as lower Renyi-2 $S_2$ and von Neumann $S_{vN}$ absolute entanglement entropy reconstruction errors. More importantly, for all tested QSK system sizes, HRBM NQS demonstrates a superior expressivity in faithfully reproducing the entire entanglement spectrum of the QSK model from the top eigenvalues down to the tail end across 15 orders of magnitude, while RBM NQS consistently overestimates the sub-dominant modes. This work furnishes a proof-of-concept demonstrating that hyperbolic non-autoregressive NQS ansatze, thanks to the exponential volume of the hyperbolic geometry underlying their constructions, might be more natural at representing volume-law quantum systems than conventional Euclidean NQS. Furthermore, an interesting byproduct of this work is the polynomial scaling result of RBM-type NQS ansatze in the QSK volume-law system as the Hilbert space dimension increases exponentially.

quant-pharXiv:2609.26032•Sep 23, 2026•0 citations

Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo

Ejaaz Merali, Mohamed Hibat-Allah, Mohammad Kohandel +2 more

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.

cond-mat.str-elCross-listedarXiv:2605.13807•May 14, 2026•0 citations

Adaptive neural quantum states: a recurrent neural network perspective

Jake McNaughton, Mohamed Hibat-Allah

Abstract Neural-network quantum states (NQS) are powerful neural-network ansätzes that have emerged as promising tools for studying quantum many-body physics through the lens of the variational principle. These architectures are known to be systematically improvable by increasing the number of parameters. Here we demonstrate an Adaptive scheme to optimize NQSs, through the example of recurrent neural networks (RNNs), using a fraction of the computation cost while reducing training fluctuations and improving the quality of variational calculations targeting ground states of prototypical models in one- and two-spatial dimensions. This Adaptive framework reduces the computational cost through training small RNNs and reusing them to initialize larger RNNs. This work opens up the possibility for optimizing graphical processing unit resources deployed in large-scale NQS simulations.

arXiv:2507.18700•Machine Learning Science and Technology•Sep 15, 2026•0 citations

Autoregressive neural quantum states of Fermi Hubbard models

Eduardo Ibarra-García-Padilla, Hannah Lange, Roger G. Melko +4 more

Neural quantum states (NQSs) have emerged as a powerful ansatz for variational quantum Monte Carlo studies of strongly correlated systems. Here, we apply recurrent neural networks (RNNs) and autoregressive transformer neural networks to the Fermi-Hubbard and the (non-Hermitian) Hatano-Nelson-Hubbard models in one and two dimensions. In both cases, we observe that the convergence of the RNN ansatz is challenged when increasing the interaction strength. We present a physically motivated and easy-to-implement strategy for improving the optimization, namely, by ramping of the model parameters. Furthermore, we investigate the advantages and disadvantages of the autoregressive sampling property of both network architectures. For the Hatano-Nelson-Hubbard model, we identify convergence issues that stem from the autoregressive sampling scheme in combination with the non-Hermitian nature of the model. Our findings provide insights into the challenges of the NQS approach and make the first step towards exploring strongly correlated electrons using this ansatz.

arXiv:2411.07144•Physical Review Research•Feb 3, 2025•9 citations

Modelling non-markovian quantum processes with recurrent neural networks

Leonardo Banchi, Edward Grant, Rocchetto, A +1 more

Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.

•Dec 1, 2018•70 citations
In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincare hyperbolic GRU, to new variants including Poincare RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing the new non-Euclidean hyperbolic NQS ansatzes, we generalize the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincare GRU NQS when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the Heisenberg J1J2J_1J_2J1​J2​ and J1J2J3J_1J_2J_3J1​J2​J3​ models. Here, using larger systems consisting of 100 spins, we find that all four hyperbolic RNN/GRU NQS variants always outperform their respective Euclidean counterpart with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different J2J_2J2​ and (J2,J3J_2, J_3J2​,J3​) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatzes are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Poincare RNN/GRU and Lorentz RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.
In the first part of this work, we construct the first type of two-dimensional (2D) hyperbolic neural quantum state (NQS) in the form of the Lorentz 2DRNN (Recurrent Neural Network) and benchmark its performance against the Euclidean 2DRNN in the paradigmatic N×NN \times NN×N 2D Transverse Field Ising Model (2DTFIM) setting with different lattice sizes up to N = 12 at different transverse magnetic field strengths. We find that hyperbolic Lorentz 2DRNN NQS robustly outperforms Euclidean 2DRNN NQS in our experiments when the system is at the phase transition point, where the physics can be described by a conformal field theory (CFT) - known via the AdS/CFT holographic duality to possess an underlying hyperbolic geometry. In the second part of this work, we benchmark the performances of recently introduced one-dimensional (1D) hyperbolic NQS ansatzes (including Poincare RNN/GRU and Lorentz RNN/GRU) against their Euclidean versions in N×NN \times NN×N 2DTFIM, converted to a one-dimensional sequence. The findings extend our previous results showing that 1D hyperbolic NQS robustly outperforms 1D Euclidean NQS, thanks to the combined effects of the induced 1st and Nth neighbor interaction hierarchy and the CFT physics at criticality. While studies on larger system sizes remain future work, this study serves as a proof-of-concept for the utility, effectiveness, and superior performance of 1D and 2D hyperbolic NQS ansatzes compared to conventional Euclidean NQS models in many-body quantum systems - particularly those exhibiting structural hierarchy, criticality, or a combination of both.
We construct the first type of non-Euclidean non-autoregressive neural quantum state (NQS) in the form of the hyperbolic Restricted Boltzmann Machine (HRBM), which is studied in the variational Monte-Carlo (VMC) setting of the Quantum Sherrington-Kirkpatrick (QSK) model whose ground state exhibits volume-law entanglement. Across a 512-fold increase in the Hilbert space dimension corresponding to a system size increase from N=14N=14N=14 to N=24N=24N=24, HRBM NQS robustly outperforms its Euclidean version, the RBM NQS, in terms of better ground state energy optimization as well as lower Renyi-2 S2S_2S2​ and von Neumann SvNS_{vN}SvN​ absolute entanglement entropy reconstruction errors. More importantly, for all tested QSK system sizes, HRBM NQS demonstrates a superior expressivity in faithfully reproducing the entire entanglement spectrum of the QSK model from the top eigenvalues down to the tail end across 15 orders of magnitude, while RBM NQS consistently overestimates the sub-dominant modes. This work furnishes a proof-of-concept demonstrating that hyperbolic non-autoregressive NQS ansatze, thanks to the exponential volume of the hyperbolic geometry underlying their constructions, might be more natural at representing volume-law quantum systems than conventional Euclidean NQS. Furthermore, an interesting byproduct of this work is the polynomial scaling result of RBM-type NQS ansatze in the QSK volume-law system as the Hilbert space dimension increases exponentially.
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 52×5252\times5252×52 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 Neural-network quantum states (NQS) are powerful neural-network ansätzes that have emerged as promising tools for studying quantum many-body physics through the lens of the variational principle. These architectures are known to be systematically improvable by increasing the number of parameters. Here we demonstrate an Adaptive scheme to optimize NQSs, through the example of recurrent neural networks (RNNs), using a fraction of the computation cost while reducing training fluctuations and improving the quality of variational calculations targeting ground states of prototypical models in one- and two-spatial dimensions. This Adaptive framework reduces the computational cost through training small RNNs and reusing them to initialize larger RNNs. This work opens up the possibility for optimizing graphical processing unit resources deployed in large-scale NQS simulations.
Neural quantum states (NQSs) have emerged as a powerful ansatz for variational quantum Monte Carlo studies of strongly correlated systems. Here, we apply recurrent neural networks (RNNs) and autoregressive transformer neural networks to the Fermi-Hubbard and the (non-Hermitian) Hatano-Nelson-Hubbard models in one and two dimensions. In both cases, we observe that the convergence of the RNN ansatz is challenged when increasing the interaction strength. We present a physically motivated and easy-to-implement strategy for improving the optimization, namely, by ramping of the model parameters. Furthermore, we investigate the advantages and disadvantages of the autoregressive sampling property of both network architectures. For the Hatano-Nelson-Hubbard model, we identify convergence issues that stem from the autoregressive sampling scheme in combination with the non-Hermitian nature of the model. Our findings provide insights into the challenges of the NQS approach and make the first step towards exploring strongly correlated electrons using this ansatz.
Quantum systems interacting with an unknown environment are notoriously difficult to model, especially in presence of non-Markovian and non-perturbative effects. Here we introduce a neural network based approach, which has the mathematical simplicity of the Gorini–Kossakowski–Sudarshan–Lindblad master equation, but is able to model non-Markovian effects in different regimes. This is achieved by using recurrent neural networks (RNNs) for defining Lindblad operators that can keep track of memory effects. Building upon this framework, we also introduce a neural network architecture that is able to reproduce the entire quantum evolution, given an initial state. As an application we study how to train these models for quantum process tomography, showing that RNNs are accurate over different times and regimes.