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

Communications Physics

Efficient computation using spatial-photonic Ising machines with low-rank and circulant matrix constraints

Richard Zhipeng Wang, James Cummins, Marvin Syed, Nikita Stroev, Georgios Pastras, Jason Sakellariou, S. I. Tsintzos, Alexis Askitopoulos, Daniele Veraldi, Marcello Calvanese Strinati, Silvia Gentilini, Davide Pierangeli, Claudio Conti, Natalia G. Berloff

Published Mar 3, 2025• 13 citations
Publisher

Abstract

Abstract Spatial-photonic Ising machines (SPIMs) have shown promise as an energy-efficient Ising machine, but currently can only solve a limited set of Ising problems. There is currently limited understanding on what experimental constraints may impact the performance of SPIM, and what computationally intensive problems can be efficiently solved by SPIM. Our results indicate that the performance of SPIMs is critically affected by the rank and precision of the coupling matrices. By developing and assessing advanced decomposition techniques, we expand the range of problems SPIMs can solve, overcoming the limitations of traditional Mattis-type matrices. Our approach accommodates a diverse array of coupling matrices, including those with inherently low ranks, applicable to complex NP-complete problems. We explore the practical benefits of the low-rank approximation in optimisation tasks, particularly in financial optimisation, to demonstrate the real-world applications of SPIMs. Finally, we evaluate the computational limitations imposed by SPIM hardware precision and suggest strategies to optimise the performance of these systems within these constraints.

DOI
10.1038/s42005-025-01987-5

Related papers

Efficient Computation Using Spatial-Photonic Ising Machines: Utilizing Low-Rank and Circulant Matrix Constraints

Richard Zhipeng Wang, James Cummins, Marvin Syed +11 more

We explore the potential of spatial-photonic Ising machines (SPIMs) to address computationally intensive Ising problems that employ low-rank and circulant coupling matrices. Our results indicate that the performance of SPIMs is critically affected by the rank and precision of the coupling matrices. By developing and assessing advanced decomposition techniques, we expand the range of problems SPIMs can solve, overcoming the limitations of traditional Mattis-type matrices. Our approach accommodates a diverse array of coupling matrices, including those with inherently low ranks, applicable to complex NP-complete problems. We explore the practical benefits of low-rank approximation in optimization tasks, particularly in financial optimization, to demonstrate the real-world applications of SPIMs. Finally, we evaluate the computational limitations imposed by SPIM hardware precision and suggest strategies to optimize the performance of these systems within these constraints.

arXiv:2406.01400•Jun 3, 2024•0 citations

Low-Rank Combinatorial Optimization and Statistical Learning by Spatial Photonic Ising Machine

Hiroshi Yamashita, Ken-ichi Okubo, Suguru Shimomura +3 more

The spatial photonic Ising machine (SPIM) [13D. Pierangeli et al., Large-Scale Photonic Ising Machine by Spatial Light Modulation, Phys. Rev. Lett. 122, 213902 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.213902] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.

arXiv:2303.14993•Physical Review Letters•Aug 7, 2023•47 citations

Encoding Arbitrary Ising Hamiltonians on Spatial Photonic Ising Machines

Jason Sakellariou, Alexis Askitopoulos, Georgios Pastras +1 more

Photonic Ising machines constitute an emergent new paradigm of computation geared toward tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial photonic Ising machines (SPIMs) have proven advantageous for simulating fully connected large-scale spin systems. Fine control of a general interaction matrix J has been accomplished so far only through matrix decomposition methods. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, allowing the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected value and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach significantly expands the applicability of SPIMs for real-world applications, as it is more efficient than matrix decomposition methods in the case of sparse problems. It paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models on SPIM devices.

Physical Review Letters•May 20, 2025•12 citations

Encoding arbitrary Ising Hamiltonians on Spatial Photonic Ising Machines

Jason Sakellariou, Alexis Askitopoulos, Georgios Pastras +1 more

Photonic Ising Machines constitute an emergent new paradigm of computation, geared towards tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial Photonic Ising Machines have proven to be advantageous for simulating fully connected large-scale spin systems. However, fine control of a general interaction matrix $J$ has so far only been accomplished through eigenvalue decomposition methods that either limit the scalability or increase the execution time of the optimization process. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, enabling the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected values and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach greatly expands the applicability of SPIMs for real-world applications without sacrificing any of the inherent advantages of the system, and paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models, on SPIM devices.

arXiv:2407.09161•Jul 12, 2024•1 citation

Convolutional formulation of large-scale quadratic unconstrained binary optimization with dense interactions

Hiroshi Yamashita, Hideyuki Suzuki

Abstract The spatial photonic Ising machine (SPIM) is a promising optical hardware solver for large-scale combinatorial optimization problems with dense interactions. As the SPIM can represent Ising problems with rank-one coupling matrices, multiplexed versions have been proposed to enhance applicability to higher-rank interactions. However, the multiplexing cost reduces implementation efficiency, and even without multiplexing, the SPIM can represent coupling matrices beyond rank-one. To clarify the intrinsic representation power of the SPIM, we propose spatial quadratic unconstrained binary optimization (spQUBO), a formulation of Ising problems with spatially convolutional structures. We prove that any spQUBO reduces to a two-dimensional spQUBO with the convolutional structure preserved, which can be efficiently implemented on the SPIM without multiplexing. We demonstrate its applicability to distance-based combinatorial optimization, including placement problems and clustering problems. These results advance our understanding of the class of optimization problems where SPIMs exhibit unique advantage in efficiency and scalability. Furthermore, the convolutional structure of spQUBO also enables efficient computation using Fast Fourier Transforms.

arXiv:2506.24008•Communications Physics•Jul 29, 2026•1 citation

Enhanced, fully connected 360 000-spin spatial photonic Ising machine

Junze Yao, Rongwei Zhu, Junjie Yu

The spatial photonic Ising machine (SPIM) is a promising architecture that leverages the scalability and parallelism of photons to solve Ising problems in combinatorial optimization. SPIMs have been held back fundamentally by limited computational accuracy. This study overcomes that limitation through a hybrid approach, integrating aberration correction with a dynamic algorithm to update simultaneously a number of flipped spins, rather than just one. The method demonstrably improves computational accuracy and presents a way for SPIMs to scale to significantly larger systems (order 10${}^{5}$ spins), a milestone on the path to practical application.

Physical Review Applied•Sep 25, 2025•5 citations
Abstract Spatial-photonic Ising machines (SPIMs) have shown promise as an energy-efficient Ising machine, but currently can only solve a limited set of Ising problems. There is currently limited understanding on what experimental constraints may impact the performance of SPIM, and what computationally intensive problems can be efficiently solved by SPIM. Our results indicate that the performance of SPIMs is critically affected by the rank and precision of the coupling matrices. By developing and assessing advanced decomposition techniques, we expand the range of problems SPIMs can solve, overcoming the limitations of traditional Mattis-type matrices. Our approach accommodates a diverse array of coupling matrices, including those with inherently low ranks, applicable to complex NP-complete problems. We explore the practical benefits of the low-rank approximation in optimisation tasks, particularly in financial optimisation, to demonstrate the real-world applications of SPIMs. Finally, we evaluate the computational limitations imposed by SPIM hardware precision and suggest strategies to optimise the performance of these systems within these constraints.
We explore the potential of spatial-photonic Ising machines (SPIMs) to address computationally intensive Ising problems that employ low-rank and circulant coupling matrices. Our results indicate that the performance of SPIMs is critically affected by the rank and precision of the coupling matrices. By developing and assessing advanced decomposition techniques, we expand the range of problems SPIMs can solve, overcoming the limitations of traditional Mattis-type matrices. Our approach accommodates a diverse array of coupling matrices, including those with inherently low ranks, applicable to complex NP-complete problems. We explore the practical benefits of low-rank approximation in optimization tasks, particularly in financial optimization, to demonstrate the real-world applications of SPIMs. Finally, we evaluate the computational limitations imposed by SPIM hardware precision and suggest strategies to optimize the performance of these systems within these constraints.
The spatial photonic Ising machine (SPIM) [13D. Pierangeli et al., Large-Scale Photonic Ising Machine by Spatial Light Modulation, Phys. Rev. Lett. 122, 213902 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.213902] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.
Photonic Ising machines constitute an emergent new paradigm of computation geared toward tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial photonic Ising machines (SPIMs) have proven advantageous for simulating fully connected large-scale spin systems. Fine control of a general interaction matrix J has been accomplished so far only through matrix decomposition methods. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, allowing the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected value and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach significantly expands the applicability of SPIMs for real-world applications, as it is more efficient than matrix decomposition methods in the case of sparse problems. It paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models on SPIM devices.
Photonic Ising Machines constitute an emergent new paradigm of computation, geared towards tackling combinatorial optimization problems that can be reduced to the problem of finding the ground state of an Ising model. Spatial Photonic Ising Machines have proven to be advantageous for simulating fully connected large-scale spin systems. However, fine control of a general interaction matrix JJJ has so far only been accomplished through eigenvalue decomposition methods that either limit the scalability or increase the execution time of the optimization process. We introduce and experimentally validate a SPIM instance that enables direct control over the full interaction matrix, enabling the encoding of Ising Hamiltonians with arbitrary couplings and connectivity. We demonstrate the conformity of the experimentally measured Ising energy with the theoretically expected values and then proceed to solve both the unweighted and weighted graph partitioning problems, showcasing a systematic convergence to an optimal solution via simulated annealing. Our approach greatly expands the applicability of SPIMs for real-world applications without sacrificing any of the inherent advantages of the system, and paves the way to encoding the full range of NP problems that are known to be equivalent to Ising models, on SPIM devices.
Abstract The spatial photonic Ising machine (SPIM) is a promising optical hardware solver for large-scale combinatorial optimization problems with dense interactions. As the SPIM can represent Ising problems with rank-one coupling matrices, multiplexed versions have been proposed to enhance applicability to higher-rank interactions. However, the multiplexing cost reduces implementation efficiency, and even without multiplexing, the SPIM can represent coupling matrices beyond rank-one. To clarify the intrinsic representation power of the SPIM, we propose spatial quadratic unconstrained binary optimization (spQUBO), a formulation of Ising problems with spatially convolutional structures. We prove that any spQUBO reduces to a two-dimensional spQUBO with the convolutional structure preserved, which can be efficiently implemented on the SPIM without multiplexing. We demonstrate its applicability to distance-based combinatorial optimization, including placement problems and clustering problems. These results advance our understanding of the class of optimization problems where SPIMs exhibit unique advantage in efficiency and scalability. Furthermore, the convolutional structure of spQUBO also enables efficient computation using Fast Fourier Transforms.
The spatial photonic Ising machine (SPIM) is a promising architecture that leverages the scalability and parallelism of photons to solve Ising problems in combinatorial optimization. SPIMs have been held back fundamentally by limited computational accuracy. This study overcomes that limitation through a hybrid approach, integrating aberration correction with a dynamic algorithm to update simultaneously a number of flipped spins, rather than just one. The method demonstrably improves computational accuracy and presents a way for SPIMs to scale to significantly larger systems (order 105{}^{5}5 spins), a milestone on the path to practical application.