Hardware Acceleration of Frustrated Lattice Systems using Convolutional Restricted Boltzmann Machine

TL;DR

Geometric frustration creates complex energy landscapes that challenge conventional Monte Carlo sampling. The authors map the classical SS Ising lattice to a Convolutional Restricted Boltzmann Machine ($CRBM$) and implement a two-stage FPGA accelerator that exploits translational symmetry to achieve efficient, parallel Gibbs sampling. The approach recovers all known SS Ising phases up to $18\times18$ and demonstrates 3–5 orders of magnitude speedup over GPU-based sampling, with sampling throughput approaching the performance of quantum annealers while offering scalability and room-temperature operation. This work provides a practical pathway to scalable, symmetry-aware hardware for large-scale simulations of quantum materials and lays the groundwork for CRBM-based variational Monte Carlo in lattice systems.

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.

Figures (6)

• Figure 1: Convolutional Restricted Boltzmann Machine Mapping: (a) The Shastry Sutherland (SS) Ising lattice structure with a $3\times3$ unit cell (b) Mapping the spin interactions to hidden nodes using the formalism in Carleo2018, resulting in a sparse RBM (c) Transforming the sparse RBM into a CRBM. Each colored convolutional filter represents a bond interaction within the 3X3 unit cell of the SS lattice. For this particular example, there are 10 bonds in the unit cell corresponding to 10 convolutional filters. (d) Gibbs sampling of the CRBM to obtain low lying energy spin configurations. The iterative cycle of updating visible and hidden neurons generates spin configurations that stochastically represent the Boltzmann distribution in Eq.\ref{['Boltzmann_eq']}.
• Figure 2: Hardware Implementation of CRBM: (a) The spatially parallelized, two-stage pipelined, and logically synthesized CRBM hardware accelerator. The two columns of operations between the node registers represent the two parallel processing pipelines. (b) Convolution operation that models the spin interactions. This is implemented using a mask-and-accumulate operation with multiplexer-based bitwise multiplication, enabling resource-efficient and high-speed sampling. (c) Zero padding ensures that the reconstructed visible neuron retains the original dimensions. (d) The wrapping module emulates periodic or open boundary conditions of the lattice.
• Figure 3: Phase Diagram and ground state spin configurations: (a) Magnetization Phase diagram of the SS lattice of size $18\times18$ obtained using the CRBM hardware accelerator. The thermodynamic limit was approximated by imposing periodic boundary conditions. (b) Average magnetization curve as a function of longitudinal field along $J_2/J_1 = 1.0$. The broadening near transition points arises from finite-temperature simulations and the limited precision of fixed-point convolutional filter representations. Additionally, the peaks in the first derivative of the curve coincide with the theoretical critical points. (c) Neel AFM spin configuration (d) Spin configuration at the critical point $J_2/J_1 = 1.0, h_z = 2.0$ and (e) Spin configuration corresponding to the 1/3-magnetization plateau. (f) Classical spin liquid state emerging from lateral confinementBrahma2024 in an SS lattice of size $12\times18$. The problem parameters were $J_2/J_1 = 1.5, h_z = 4.5$. The blue squares represent -1 and the green squares represent 1.
• Figure 4: Static Structure factor of various spin configurations: (a) Static structure factor of the spin configuration at the critical transition between Neel AFM and 1/3 fractional plateau. The structure factor appears as a broadened superposition of peaks characteristic of both the AFM and fractional plateau phases. (b) Structure factor of the classical spin liquid phase at $J_2/J_1 = 1.5, h_z = 4.5$. (c) A cut along the high symmetry points of (a) illustrates the transition of structure factor from Neel AFM to 1/3 fractional plateau. At the critical point, the peaks broaden over a range of momenta depicting the destruction of long-range order-a signature of critical phase transitions. (d) A cut along the high symmetry points of (b), showing the transition from an ordered state at $J_2/J_1 = 1.5, h_z = 2.5$ to a classical spin liquid phase, with the transition occuring at $h_z = 3.5$. At the transition point and within the classical spin liquid phase, $S(\vec{q})$ exhibits broadening over a wide range of momenta, reflecting macroscopic degeneracy—a defining characteristic of classical spin liquids Ramirez1994.
• Figure 5: Hardware Acceleration across various thermodynamic magnetic phases: (a) TTS Scaling of the hardware across different thermodynamic magnetization phases. The FM ($J_2/J_1 = 1.0, h_z = 7.0$) and the dimer phase ($J_2/J_1 = 2.5, h_z = 0.0$) are the easiest to sample due to the presence of large longitudinal bias and macroscopic degeneracy, respectively. In contrast, Neel AFM ($J_2/J_1 = 1.0, h_z = 0.0$) and 1/3 fractional ($J_2/J_1 = 0.5, h_z = 4.0$) are much more challenging to sample due to the finite degeneracy of ground state solutions and the presence of long-range magnetic order. (b) Hardware acceleration of our digital implementation compared to GPU-based CRBM sampling algorithm.