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Optics Express

Combinatorial clustering with a coherent XY machine

Kyungduk Kim, Masahito Kumagai, Y. Yamamoto

Published Aug 20, 2024• 4 citations
Publisher

Abstract

We propose and numerically study the performance of an all-optical machine for tackling combinatorial clustering, one of the unsupervised machine learning problems. A problem instance is encoded into the phase of time-multiplexed optical pulses, which are coupled repulsively through optical delay lines. To maintain uniform pulse amplitudes, we utilize a nonlinear amplifier with gain saturation, enabling our optical architecture to emulate the classical XY-spin system. This solver, called the coherent XY machine, leads to the formation of clusters in an optical phase space and allows us to efficiently identify the solution with post-processing. Additionally, we implement momentum in our solver to provide a powerful mechanism for escaping local minima and searching for the global optimum. Benchmarking our approach with a most advanced Ising-spin-based solver reveals a two-orders-of-magnitude improvement in the time-to-solution of the algorithm. Furthermore, our approach exhibits scaling advantages for larger problems, which will facilitate time- and energy-efficient data clustering.

DOI
10.1364/oe.526050

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We propose and numerically study the performance of an all-optical machine for tackling combinatorial clustering, one of the unsupervised machine learning problems. A problem instance is encoded into the phase of time-multiplexed optical pulses, which are coupled repulsively through optical delay lines. To maintain uniform pulse amplitudes, we utilize a nonlinear amplifier with gain saturation, enabling our optical architecture to emulate the classical XY-spin system. This solver, called the coherent XY machine, leads to the formation of clusters in an optical phase space and allows us to efficiently identify the solution with post-processing. Additionally, we implement momentum in our solver to provide a powerful mechanism for escaping local minima and searching for the global optimum. Benchmarking our approach with a most advanced Ising-spin-based solver reveals a two-orders-of-magnitude improvement in the time-to-solution of the algorithm. Furthermore, our approach exhibits scaling advantages for larger problems, which will facilitate time- and energy-efficient data clustering.
Many tasks in our modern life, such as planning an efficient travel, image processing and optimizing integrated circuit design, are modeled as complex combinatorial optimization problems with binary variables. Such problems can be mapped to finding a ground state of the Ising Hamiltonian, thus various physical systems have been studied to emulate and solve this Ising problem. Recently, networks of mutually injected optical oscillators, called coherent Ising machines, have been developed as promising solvers for the problem, benefiting from programmability, scalability and room temperature operation. Here, we report a 16-bit coherent Ising machine based on a network of time-division-multiplexed femtosecond degenerate optical parametric oscillators. The system experimentally gives more than 99.6% of success rates for one-dimensional Ising ring and nondeterministic polynomial-time (NP) hard instances. The experimental and numerical results indicate that gradual pumping of the network combined with multiple spectral and temporal modes of the femtosecond pulses can improve the computational performance of the Ising machine, offering a new path for tackling larger and more complex instances.
We numerically demonstrate that introducing XY-spin dynamics enabled by phase-insensitive optical parametric gain enhances the performance of coherent Ising machines (CIMs). Whereas conventional CIMs based on phase-sensitive optical parametric gain confine spins to two discrete Ising states, nondegenerate optical parametric oscillators with phase-insensitive gain support continuous-phase XY spins, facilitating spin flips that enable the system to escape local minima. Benchmarking on Wishart-planted problem instances shows that gradually transitioning from XY to Ising spins reduces the time-to-solution by roughly an order of magnitude relative to purely Ising-based dynamics. Furthermore, tailoring the transitions between XY-like and Ising-like regimes yields additional improvements beyond a simple unidirectional XY-to-Ising transition. Our results establish a new framework for engineering CIM dynamics in full phase-quadrature space and point toward fully optical architectures for efficient combinatorial optimization.
Coherent Ising machine (CIM) is a hardware solver that simulates the Ising model and finds optimal solutions to combinatorial optimization problems. However, for practical tasks, the computational process may be trapped in local minima, which is a key challenge for CIM. In this work, we design a CIM structure with a spiking neural network by adding dissipative pulses, which are anti-symmetrically coupled to the degenerate optical parametric oscillator pulses in CIM with a measurement feedback system. We find that the unstable oscillatory region of the spiking neural network could assist the CIM to escape from the trapped local minima. Moreover, we show that the machine has a different search mechanism than CIM, which can achieve a higher solution success probability and speed-up effect.
ABSTRACT Many real‐world applications can be categorized as combinatorial optimization problems (COPs), yet their inherent NP‐hardness and exponential computational complexity pose significant barriers to efficient solutions. Designed specifically for solving COPs, the Ising machine (IM) demonstrates superior performance over traditional Von Neumann‐based computing architectures. Among different IM implementations, optical IMs (OIMs) are especially promising due to their inherent high‐speed and energy efficiency properties. Nevertheless, current OIMs are constrained by serial processing architectures, posing significant challenges for solving large‐scale COPs. To address this challenge, we propose and experimentally demonstrate a multiplexed parallel OIM architecture spanning the optical and electrical domains. The system solved a square lattice MAX‐CUT instance with over 100,000 spins and achieved a fourfold speedup with four‐channel parallelism. In addition, channel‐wise calibration mitigated device mismatch and aligned the nonlinear responses across channels, enabling the four‐channel system to maintain solution quality comparable to the single‐channel system across small benchmark problems of varying difficulty. We also explored a method combining hierarchical clustering and noise injection, which identified the known optimal route within the clustering‐constrained search space of the 14‐city TSP. These results demonstrate the potential of the proposed architecture for larger and more complex optimization problems.
Time-multiplexed networks of degenerate optical parametric oscillators have demonstrated remarkable success in simulating coupled Ising spins, thus providing a promising route to solving complex combinatorial optimization problems. In these systems $\unicode{x2014}$ referred to as coherent Ising machines $\unicode{x2014}$ the spins are encoded in the phases of the oscillators, and spin states are measured at the system output using phase-sensitive techniques. Here, we present an experimental demonstration of a conceptually new optical Ising machine based upon spontaneous polarization symmetry breaking within a coherently-driven optical fibre Kerr nonlinear resonator. In our scheme, the spin states are encoded using polarization, which allows the state of the network to be robustly read out using straightforward intensity measurements. Furthermore, by operating within a recently-discovered regime where the interplay between nonlinearity and topology fundamentally safeguards the system's symmetry, we ensure that our spins evolve without unwanted biases. This enables continuous Ising machine trials at optical data rates for up to an hour without resetting or manual adjustments. With an all-fibre implementation that relies solely on standard telecommunications components, we believe our work paves the way for substantial advances in the performance and stability of coherent optical Ising machines for applications ranging from financial modeling to drug discovery and machine learning.
Finding the ground state of Ising spin glasses is notoriously difficult due to disorder and frustration. Often, this challenge is framed as a combinatorial optimization problem, for which a common strategy employs simulated annealing, a Monte Carlo (MC)-based algorithm that updates spins one at a time. Yet, these localized updates can cause the system to become trapped in local minima. Cluster algorithms (CAs) were developed to address this limitation and have demonstrated considerable success in studying ferromagnetic systems; however, they tend to encounter percolation issues when applied to generic spin glasses. In this work, we introduce a novel CA designed to tackle these challenges by leveraging precomputed two-point correlations, aiming solve combinatorial optimization problems in the form of Max-Cut more efficiently. In our approach, clusters are formed probabilistically based on these correlations. Various classical and quantum algorithms can be employed to generate correlations that embody information about the energy landscape of the problem. By utilizing this information, the algorithm aims to identify groups of spins whose simultaneous flipping induces large transitions in configuration space with high acceptance probability - even at low energy levels - thereby escaping local minima more effectively. Notably, clusters generated using correlations from the Quantum Approximate Optimization Algorithm exhibit high acceptance rates at low temperatures. These acceptance rates often increase with circuit depth, accelerating the algorithm and enabling more efficient exploration of the solution space.