Spatial-Photonic Ising Machines and the Next Era of Computing

Table of Links

I. Introduction

II. Spim Performance, Advantages and Generality

III. Inherently Low Rank Problems

A. Properties of Low Rank Graphs

B. Weakly NP-Complete Problems and Hardware Precision Limitation

C. Limitation of Low Rank Matrix Mapping

IV. Low Rank Approximation

A. Decomposition of Target Coupling Matrix

B. How Fields Influence Ran

C. Low Rank Approximation of Coupling Matrices

D. Low-Rank Approximation of Random Coupling Matrices

E. Low Rank Approximation for Portfolio Optimization

F. Low-Rank Matrices in Restricted Boltzmann Machines

V. Constrained Number Partitioning Problem

A. Definition and Characteristics of the Constrained Number Partitioning Problem

B. Computational Hardness of Random CNP Instances

VI. Translation Invariant Problems

A. “Realistic” Spin Glass

B. Circulant Graphs

VII. Conclusions, Acknowledgements, and References

VII. CONCLUSIONS

SPIMs are emerging physical computing platforms with distinct strengths and practical constraints, setting them apart from conventional digital computing technologies. As advancements in engineering and materials technology continue, these platforms are expected to see enhanced capabilities. It is, however, imperative to identify problems and methods that can effectively utilize these unique strengths, providing a robust basis for benchmarking their performance. This paper identifies several classes of problems that are particularly well-suited for SPIM hardware. SPIMs are shown to efficiently address practical problems such as portfolio optimization through low-rank approximation techniques. This methodology also presents promising opportunities for further research, including the potential implementation of low-rank restricted Boltzmann machines on SPIMs. Furthermore, the constrained number partitioning (CNP) problem, a variation of the classic number partitioning problem, serves as a valuable benchmark for comparing the performance of SPIMs with that of classical computers. The analytically solvable circulant graph provides insights into the differences in performance between gradient-based algorithms, prevalent in many current Ising machines, and sampling-based algorithms that can be implemented on SPIMs. Additionally, SPIMs have the potential to realize many “realistic” spin glasses, extensively studied within the realm of statistical mechanics, thereby making numerous theoretical models experimentally viable.

\ Our study also highlights the importance of precision and rank in relation to the constraints of SPIM hardware. While low-rank approximations can render problems more manageable on SPIMs, the precision required for these approximations can impact computational efficiency and the accuracy of solutions. Therefore, future research must explore methods to optimize the balance between rank and precision. Beyond portfolio optimization and Boltzmann machines, SPIMs demonstrate potential in solving various NP-hard problems through innovative mapping techniques. Advanced decomposition methods, such as singular value decomposition (SVD), enable SPIMs to manage more complex coupling matrices, expanding their applicability across different optimization tasks.

\ In conclusion, SPIMs represent a promising advancement in computational technologies. By focusing on lowrank approximations, constrained number partitioning, and the implementation of sophisticated algorithms, this paper sets the stage for future investigations into the capabilities and applications of SPIMs. Continued research and development in this area are crucial for fully realizing the potential of SPIMs, paving the way for novel solutions to some of the most challenging computational problems. The broader implications of this research extend to fields such as finance, logistics, and data science, where SPIMs could significantly enhance performance and efficiency, leading to substantial advancements.

ACKNOWLEDGEMENTS

R.Z.W. M.S. G.P. J.S. A.A. S.T. C.C. D.P. S.G. M.C.S. D.V. and N.G.B. acknowledge support from HORIZON EIC-2022-PATHFINDERCHALLENGES-01 HEISINGBERG Project 101114978. R.Z.W. and N.G.B. acknowledge the support from the Julian Schwinger Foundation Grant No. JSF-19-02-0005. J.S.C. acknowledges the PhD support from the EPSRC. N.S. and N.G.B. acknowledge support from Weizmann-UK Make Connection Grant 142568. N.S. acknowledges the support provided by the Clore Fellowship. A.A. and S.T. acknowledge support from HoloCIM (CODEVELOP-ICTHEALTH/0322/0047).

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:::info Authors:

(1) Richard Zhipeng Wang, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom;

(2) James S. Cummins, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom;

(3) Marvin Syed, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom;

(4) Nikita Stroev, Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel;

(5) George Pastras, QUBITECH, Thessalias 8, Chalandri, GR 15231 Athens, Greece;

(6) Jason Sakellariou, QUBITECH, Thessalias 8, Chalandri, GR 15231 Athens, Greece;

(7) Symeon Tsintzos, QUBITECH, Thessalias 8, Chalandri, GR 15231 Athens, Greece and UBITECH ltd, 95B Archiepiskopou Makariou, CY 3020 Limassol, Cyprus;

(8) Alexis Askitopoulos, QUBITECH, Thessalias 8, Chalandri, GR 15231 Athens, Greece and UBITECH ltd, 95B Archiepiskopou Makariou, CY 3020 Limassol, Cyprus;

(9) Daniele Veraldi, Department of Physics, University Sapienza, Piazzale Aldo Moro 5, Rome 00185, Italy;

(10) Marcello Calvanese Strinati, Research Center Enrico Fermi, Via Panisperna 89A, 00185 Rome, Italy;

(11) Silvia Gentilini, Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome, Italy;

(12) Calvanese Strinati, Research Center Enrico Fermi, Via Panisperna 89A, 00185 Rome, Italy

(13) Davide Pierangeli, Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185 Rome, Italy;

(14) Claudio Conti, Department of Physics, University Sapienza, Piazzale Aldo Moro 5, Rome 00185, Italy;

(15) Natalia G. Berlof, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom ([email protected]).

:::

:::info This paper is available on arxiv under CC BY 4.0 DEED license.

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