© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2511.20911

Hardware Acceleration of Frustrated Lattice Systems using Convolutional Restricted Boltzmann Machine

Pratik Prabhanjan Brahma, J. H. Han, Tamzid Razzaque, Patel, Saavan, Sayeef Salahuddin

Published Nov 25, 2025• 0 citations

Abstract

Geometric frustration gives rise to emergent quantum phenomena and exotic phases of matter. While Monte Carlo methods are traditionally used to simulate such systems, their sampling efficiency is limited by the complexity of interactions and ground-state properties. Restricted Boltzmann Machines (RBMs), a class of probabilistic neural networks, offer improved sampling by incorporating machine learning techniques. However, fully-connected bipartite RBMs are inefficient for representing physical lattices with sparse interactions. To address this, we implement Convolutional Restricted Boltzmann Machines (CRBMs) that leverage translational symmetry inherent to lattices. Using the classical Shastry-Sutherland (SS) Ising lattice, we demonstrate (i) CRBM formulation that captures SS interactions, and (ii) digital hardware accelerator to enhance sampling performance. We simulate lattices with up to 324 spins, recovering all known phases of the SS Ising model, including the long range ordered fractional plateau. Our hardware characterizes spin behavior at critical points and within spin liquid phases. This implementation achieves a speedup of 3 to 5 orders of magnitude (33 ns to 120 ms) over GPU-based implementations. Moreover, the time-to-solution is within two orders of magnitude of quantum annealers, while offering superior scalability, room-temperature operation and reprogrammability. This work paves a pathway for scalable digital hardware that embeds physical symmetries to enable large scale simulations of material systems.

DOI: 10.48550/arxiv.2511.20911

Related papers

Convolutional restricted Boltzmann machine aided Monte Carlo: An application to Ising and Kitaev models

Daniel Alcalde Puente, Ilya Eremin

Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the convolutional restricted Boltzmann machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two dimensions.

arXiv:2008.05401•Physical review. B./Physical review. B•Nov 30, 2020•14 citations

Convolutional RBM aided Monte Carlo: an application to Ising and Kitaev models

Daniel Alcalde Puente, Ilya Eremin

Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann Machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the Convolutional Restricted Boltzmann Machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two-dimensions.

•Aug 12, 2020•0 citations

Ising Model Optimization Problems on a FPGA Accelerated Restricted Boltzmann Machine

Saavan Patel, Lili Chen, Philip Canoza +1 more

Abstract In this work we demonstrate usage of the Restricted Boltzmann Machine (RBM) as a stochastic neural network capable of solving NP-Hard Combinatorial Optimization problems efficiently. By mapping the RBM onto a reconfigurable Field Programmable Gate Array (FPGA), we can effectively hardware accelerate the RBM's stochastic sampling algorithm. We benchmark the RBM against the DWave 2000Q Quantum Adiabatic Computer and the Optical Coherent Ising Machine on two such optimization problems: the MAX-CUT problem and the Sherrington-Kirkpatrick (SK) spin glass. The hardware accelerated RBM shows asymptotic scaling either similar or better than these other accelerators. This leads to 10 7 x and 10 5 x time to solution improvement compared to the DWave 2000Q on the MAX-CUT and SK problems respectively, along with a 150x and 1000x improvement compared to the Coherent Ising Machine annealer on those problems. By utilizing commodity hardware running at room temperature, the RBM shows potential for immediate and scalable use.

arXiv:2008.04436•Dec 15, 2020•23 citations

Probabilistic Computers for Neural Quantum States

Shuvro Chowdhury, Jasper Pieterse, Navid Anjum Aadit +3 more

Neural quantum states efficiently represent many-body wavefunctions with neural networks, but the cost of Monte Carlo sampling limits their scaling to large system sizes. Here we address this challenge by combining sparse Boltzmann machine architectures with probabilistic computing hardware. We implement a probabilistic computer on field-programmable gate arrays (FPGAs) and use it as a fast sampler for energy-based neural quantum states. For the two-dimensional transverse-field Ising model at criticality, we obtain accurate ground-state energies for lattices up to 80$\times$80 (6400 spins) using a custom multi-FPGA cluster. Furthermore, we introduce a dual-sampling algorithm to train deep Boltzmann machines, replacing intractable marginalization with conditional sampling over auxiliary layers. This enables the training of sparse deep models and improves parameter efficiency relative to shallow networks. We further implement this algorithm on a single FPGA, demonstrating the training of deep Boltzmann machines for systems as large as $30 \times 30$ (900 spins). Together, these results demonstrate that probabilistic hardware can overcome the sampling bottleneck in variational simulation of quantum many-body systems, opening a path to larger system sizes and deeper variational architectures.

quant-pharXiv:2512.24558•Jan 1, 2026•0 citations

Variational Neural Network Quantum States for Frustrated Magnetic Systems

Matěj Mezera

We investigate the Shastry-Sutherland model (SSM), i.e., spin-1/2 quantum Heisen- berg model on a Shastry-Sutherland lattice, using a newly emerging approach exploiting well-developed machine learning techniques. We utilize neural networks as variational quantum states in quantum Monte Carlo investigations of the ground state properties. We first focus on SSM without an external magnetic field. For small lattices accessible via exact diagonalization, we compare the precision of various architectures based on re- stricted Boltzmann machines (RBM) or group-convolutional neural networks. The most versatile and precise architecture, namely complex-valued RBM, is then applied for larger lattices. Here we investigate the frustrated regime. We show that the RBM is able to represent all three of the major and fundamentally different phases of SSM. Finally, we apply the complex-valued RBM for SSM in a finite external magnetic field. We find that it cannot capture the intriguing steps in magnetization typical for SSM correctly due to its tendency to prefer more ordered states with higher magnetization. 1

•Jan 1, 2022•0 citations

Simulating the Shastry-Sutherland Ising Model Using Quantum Annealing

Paul Kairys, Andrew D. King, Isil Ozfidan +4 more

A core concept in condensed matter physics is geometric frustration that leads to emergent spin phases in magnetic materials.These distinct phases, which depart from the conventional ferromagnet or the antiferromagnet, require unique computational techniques to decipher.In this study, we use the canonical Ising Shastry-Sutherland lattice to demonstrate new techniques for solving frustrated Hamiltonians using a quantum annealer of programmable superconducting qubits.This Hamiltonian can be tuned to produce a variety of intriguing ground states ranging from short-and long-range orders and fractional order parameters.We show that a large-scale finite-field quantum annealing experiment is possible on 468 logical spins of this model embedded into the quantum hardware.We determine microscopic spin configurations using an iterative quantum annealing protocol and develop mean-field boundary conditions to attenuate finitesize effects and defects.We not only recover all phases of the Shastry-Sutherland Ising model-including the well-known fractional magnetization plateau in a longitudinal field-but also predict the spin behavior at the critical points with significant ground-state degeneracy and in the presence of defects.The results lead us to establish the connection to the diffuse neutron scattering experiments by calculation of the static structure factors.

arXiv:2003.01019•PRX Quantum•Dec 14, 2020•62 citations

Geometric frustration gives rise to emergent quantum phenomena and exotic phases of matter. While Monte Carlo methods are traditionally used to simulate such systems, their sampling efficiency is limited by the complexity of interactions and ground-state properties. Restricted Boltzmann Machines (RBMs), a class of probabilistic neural networks, offer improved sampling by incorporating machine learning techniques. However, fully-connected bipartite RBMs are inefficient for representing physical lattices with sparse interactions. To address this, we implement Convolutional Restricted Boltzmann Machines (CRBMs) that leverage translational symmetry inherent to lattices. Using the classical Shastry-Sutherland (SS) Ising lattice, we demonstrate (i) CRBM formulation that captures SS interactions, and (ii) digital hardware accelerator to enhance sampling performance. We simulate lattices with up to 324 spins, recovering all known phases of the SS Ising model, including the long range ordered fractional plateau. Our hardware characterizes spin behavior at critical points and within spin liquid phases. This implementation achieves a speedup of 3 to 5 orders of magnitude (33 ns to 120 ms) over GPU-based implementations. Moreover, the time-to-solution is within two orders of magnitude of quantum annealers, while offering superior scalability, room-temperature operation and reprogrammability. This work paves a pathway for scalable digital hardware that embeds physical symmetries to enable large scale simulations of material systems.

Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the convolutional restricted Boltzmann machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two dimensions.

Machine learning is becoming widely used in analyzing the thermodynamics of many-body condensed matter systems. Restricted Boltzmann Machine (RBM) aided Monte Carlo simulations have sparked interest recently, as they manage to speed up classical Monte Carlo simulations. Here we employ the Convolutional Restricted Boltzmann Machine (CRBM) method and show that its use helps to reduce the number of parameters to be learned drastically by taking advantage of translation invariance. Furthermore, we show that it is possible to train the CRBM at smaller lattice sizes, and apply it to larger lattice sizes. To demonstrate the efficiency of CRBM we apply it to the paradigmatic Ising and Kitaev models in two-dimensions.

Abstract In this work we demonstrate usage of the Restricted Boltzmann Machine (RBM) as a stochastic neural network capable of solving NP-Hard Combinatorial Optimization problems efficiently. By mapping the RBM onto a reconfigurable Field Programmable Gate Array (FPGA), we can effectively hardware accelerate the RBM's stochastic sampling algorithm. We benchmark the RBM against the DWave 2000Q Quantum Adiabatic Computer and the Optical Coherent Ising Machine on two such optimization problems: the MAX-CUT problem and the Sherrington-Kirkpatrick (SK) spin glass. The hardware accelerated RBM shows asymptotic scaling either similar or better than these other accelerators. This leads to 10 7 x and 10 5 x time to solution improvement compared to the DWave 2000Q on the MAX-CUT and SK problems respectively, along with a 150x and 1000x improvement compared to the Coherent Ising Machine annealer on those problems. By utilizing commodity hardware running at room temperature, the RBM shows potential for immediate and scalable use.

Neural quantum states efficiently represent many-body wavefunctions with neural networks, but the cost of Monte Carlo sampling limits their scaling to large system sizes. Here we address this challenge by combining sparse Boltzmann machine architectures with probabilistic computing hardware. We implement a probabilistic computer on field-programmable gate arrays (FPGAs) and use it as a fast sampler for energy-based neural quantum states. For the two-dimensional transverse-field Ising model at criticality, we obtain accurate ground-state energies for lattices up to 80$\times$80 (6400 spins) using a custom multi-FPGA cluster. Furthermore, we introduce a dual-sampling algorithm to train deep Boltzmann machines, replacing intractable marginalization with conditional sampling over auxiliary layers. This enables the training of sparse deep models and improves parameter efficiency relative to shallow networks. We further implement this algorithm on a single FPGA, demonstrating the training of deep Boltzmann machines for systems as large as

30 \times 30

(900 spins). Together, these results demonstrate that probabilistic hardware can overcome the sampling bottleneck in variational simulation of quantum many-body systems, opening a path to larger system sizes and deeper variational architectures.

We investigate the Shastry-Sutherland model (SSM), i.e., spin-1/2 quantum Heisen- berg model on a Shastry-Sutherland lattice, using a newly emerging approach exploiting well-developed machine learning techniques. We utilize neural networks as variational quantum states in quantum Monte Carlo investigations of the ground state properties. We first focus on SSM without an external magnetic field. For small lattices accessible via exact diagonalization, we compare the precision of various architectures based on re- stricted Boltzmann machines (RBM) or group-convolutional neural networks. The most versatile and precise architecture, namely complex-valued RBM, is then applied for larger lattices. Here we investigate the frustrated regime. We show that the RBM is able to represent all three of the major and fundamentally different phases of SSM. Finally, we apply the complex-valued RBM for SSM in a finite external magnetic field. We find that it cannot capture the intriguing steps in magnetization typical for SSM correctly due to its tendency to prefer more ordered states with higher magnetization. 1

A core concept in condensed matter physics is geometric frustration that leads to emergent spin phases in magnetic materials.These distinct phases, which depart from the conventional ferromagnet or the antiferromagnet, require unique computational techniques to decipher.In this study, we use the canonical Ising Shastry-Sutherland lattice to demonstrate new techniques for solving frustrated Hamiltonians using a quantum annealer of programmable superconducting qubits.This Hamiltonian can be tuned to produce a variety of intriguing ground states ranging from short-and long-range orders and fractional order parameters.We show that a large-scale finite-field quantum annealing experiment is possible on 468 logical spins of this model embedded into the quantum hardware.We determine microscopic spin configurations using an iterative quantum annealing protocol and develop mean-field boundary conditions to attenuate finitesize effects and defects.We not only recover all phases of the Shastry-Sutherland Ising model-including the well-known fractional magnetization plateau in a longitudinal field-but also predict the spin behavior at the critical points with significant ground-state degeneracy and in the presence of defects.The results lead us to establish the connection to the diffuse neutron scattering experiments by calculation of the static structure factors.