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.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:2512.24558Jan 1, 20260 citations

Probabilistic algorithms for Ising machines from optimization to generative AI

Corentin Delacour, Xiuqi Zhang, Abdelrahman S. Abdelrahman +4 more

arXiv:2610.03972Nature Reviews Electrical EngineeringOct 15, 20260 citations

Special-Purpose Computers for the Random Ising Model

A. L. Talapov, V. B. Andreǐchenko, Виктор Степанович Доценко +1 more

Springer proceedings in physicsJan 1, 19932 citations

Quantum technology to expand soft computing

Paul J. Werbos

Soft computing was founded to unify and advance computing methods beyond the limits of binary variables, Boolean logic and Turing machines . David Deutsch invented a way to use massive quantum parallelism to massively improve the power of Turing machines in some applications. Here we give a roadmap to use quantum parallelism to massively improve the power of soft computing, focusing on tasks in optimization ranging from function optimization , finding a needle in a haystack, optimal control , energy technology and reinforcement learning (Quantum artificial general intelligence, AGI) in a path leading up to optimization of physical plants themselves put into quantum superpositions . The path begins with design and test of prototypes built on platforms such as Nuclear Magnetic Resonance , sparse quantum optics , superconductors (sQuID), quantum dots and new high speed electronics.Soft computing was founded to unify and advance computing methods beyond the limits of binary variables, Boolean logic and Turing machines . David Deutsch invented a way to use massive quantum parallelism to massively improve the power of Turing machines in some applications. Here we give a roadmap to use quantum parallelism to massively improve the power of soft computing, focusing on tasks in optimization ranging from function optimization , finding a needle in a haystack, optimal control , energy technology and reinforcement learning (Quantum artificial general intelligence, AGI) in a path leading up to optimization of physical plants themselves put into quantum superpositions . The path begins with design and test of prototypes built on platforms such as Nuclear Magnetic Resonance , sparse quantum optics , superconductors (sQuID), quantum dots and new high speed electronics.

Systems and Soft ComputingFeb 7, 202210 citations

Wavelength‐Division Multiplexing Scheme Enabling Sparse Spin Interactions For Spatial Photonic Ising Machines

Wanyi Zhang, Li Luo, Yuan Zheng +1 more

ABSTRACT Spatial photonic Ising machines (SPIMs) offer a promising physical computing architecture for solving NP‐hard combinatorial optimization problems. However, when applied to models with sparse spin interactions, conventional SPIMs still employ a fully connected scheme, which demands digital preprocessing to decompose the interaction matrix and encode full spin interactions, leading to massive resource consumption on spatial light modulators (SLMs) when the spin size is large. Here, we propose and experimentally demonstrate a wavelength‐division multiplexing scheme to enable sparse spin interactions for SPIMs, where only the non‐zero spin interactions are encoded by SLMs, bypassing the need for preprocessing computation. Leveraging the non‐Von Neumann architecture of photonic computing, this method reduces the computational complexity of a single‐spin update in Metropolis–Hastings sampling from to . We successfully applied this system to solve the Max‐Cut problem across various sparse graphs, achieving a considerable probability of finding optimal or near‐optimal solutions, with performance comparable to classical simulated annealing. The proposed spatial encoding scheme requires linear resources scaling with the number of connections between spins, providing a highly scalable and energy‐efficient optical accelerator platform for large‐scale sparse combinatorial optimization.ABSTRACT Spatial photonic Ising machines (SPIMs) offer a promising physical computing architecture for solving NP‐hard combinatorial optimization problems. However, when applied to models with sparse spin interactions, conventional SPIMs still employ a fully connected scheme, which demands digital preprocessing to decompose the interaction matrix and encode full spin interactions, leading to massive resource consumption on spatial light modulators (SLMs) when the spin size is large. Here, we propose and experimentally demonstrate a wavelength‐division multiplexing scheme to enable sparse spin interactions for SPIMs, where only the non‐zero spin interactions are encoded by SLMs, bypassing the need for preprocessing computation. Leveraging the non‐Von Neumann architecture of photonic computing, this method reduces the computational complexity of a single‐spin update in Metropolis–Hastings sampling from to . We successfully applied this system to solve the Max‐Cut problem across various sparse graphs, achieving a considerable probability of finding optimal or near‐optimal solutions, with performance comparable to classical simulated annealing. The proposed spatial encoding scheme requires linear resources scaling with the number of connections between spins, providing a highly scalable and energy‐efficient optical accelerator platform for large‐scale sparse combinatorial optimization.

Laser & Photonics ReviewSep 30, 20260 citations

Combinatorial optimization with coherent Ising machine

Takahiro Inagaki

The Japan Society of Applied PhysicsJul 10, 20180 citations

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.