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arXiv:2110.01616

A spatial-photonic Ising machine to solve the two-way\n number-partitioning problem

Vikram Ramesh, Vighnesh Natarajan, Anil Prabhakar

Published Oct 3, 2021• 2 citations
arXivHTMLPDF

Abstract

We evaluate the performance of different algorithms in minimizing the Hamiltonian of a spatial-photonic Ising machine (SPIM). We then encode the number-partitioning problem on the SPIM and adiabatically arrive at good solutions for the problem for over 16000 spins, with a time complexity that only scales linearly with problem size. Finally, we benchmark our machine performance against the classical solver, Gurobi, and also a D-Wave 5000+ quantum annealer. With just one spatial light modulator, and and adiabatic evolution scheme for the phase, our results surpass current state-of-the-art SPIMs. We reduce hardware costs, and can solve larger problems more efficiently.

DOI
10.48550/arxiv.2110.01616

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We evaluate the performance of different algorithms in minimizing the Hamiltonian of a spatial-photonic Ising machine (SPIM). We then encode the number-partitioning problem on the SPIM and adiabatically arrive at good solutions for the problem for over 16000 spins, with a time complexity that only scales linearly with problem size. Finally, we benchmark our machine performance against the classical solver, Gurobi, and also a D-Wave 5000+ quantum annealer. With just one spatial light modulator, and and adiabatic evolution scheme for the phase, our results surpass current state-of-the-art SPIMs. We reduce hardware costs, and can solve larger problems more efficiently.
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Combinatorial optimization problems are challenging for classical computing architectures, motivating the development of specialized hardware like Ising machines. The Spatial Photonic Ising Machine (SPIM) and its enhanced versions have shown promise in solving such problems by mapping them to the Ising model and leveraging optical parallelism to compute Hamiltonian energies efficiently. However, the 0-1 knapsack problem, as a representative combinatorial optimization problem, has only been reported to be solved effectively in small-scale SPIM. In this work, we analyze the factors that limit the spin scalability and propose a novel hardware-aware Hamiltonian mapping scheme to overcome this limitation. Based on this enhanced SPIM, the scheme is validated through simulated annealing experiments on a 102400-spin 0-1 knapsack problem, achieving high solution accuracy. Our method not only enables efficient solutions to large-scale 0-1 knapsack problems but also offers generalizability to other combinatorial optimization problems with similar coupling configurations. This work demonstrates the potential of SPIMs as scalable and adaptable solvers for real-world optimization challenges.
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.
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.
The spatial photonic Ising machine (SPIM) is a simple and elegant architecture capable of expanding the spin scale with ease, and thus this optical architecture is attractive for its potential in solving large-scale combinatorial optimization problems. Here, we propose a scheme for aberration compensation to improve the computing accuracy of the SPIM based on a spatial light modulator (SLM), resulting in good scalability. First, the gamma curve of the SLM is corrected with the help of double-hole interference fringes. Then, an enhanced phase shift interferometry is used to measure the aberration of the entire optical system, and this aberration is compensated by loading the compensation phase on the SLM. Moreover, an iterative algorithm using self-adaptive cluster flipping is proposed for accelerating the convergence of SPIM. During the converging process, the number of flipped spins within the cluster is updated dynamically according to the convergence speed and accuracy, which makes it possible to achieve a good balance between high speed and high accuracy. Further, a proof-of-principle experimental set-up of the optimized SPIM is implemented, and the ground-state search of an antiferromagnetic Ising model is demonstrated. As an example, this optimized Ising machine is used for solving the number partitioning problem. It is shown that the partition of 40,000 random numbers is well demonstrated, and the final partitioning accuracy is as high as 99.88%.