Category: Ising machine

Optical Spin Glasses

https://opg.optica.org/aop/abstract.cfm?URI=aop-18-2-421

Spin-glass theory emerged in the 1980s as a merger between theoretical physics and condensed matter. Soon, physicists realized that spin glasses serve as a paradigm for complex systems, as underscored by the 2021 Nobel Prize in Physics, and for applications in machine learning and neuroscience, with a profound connection with the Hopfield model and Boltzmann machines, subjects of the 2024 Nobel Prize in Physics. However, the connection with optics and photonics is even more profound and fundamental; this connection was identified as early as 1982, with the first realizations of optical neural networks. Thirty years later, the first experimental demonstration of a pillar of spin-glass theory, the replica symmetry breaking, was reported in photonics. Nowadays, many scientists consider photonics as an effective solution for new hardware in artificial intelligence, capable of reducing energy consumption in training large machine-learning modules, and also more suitable for realizing fully connected models that underpin modern data-driven analysis. The substantial equivalence between linear optical propagation and a system of interacting binary spins is now well recognized, triggering the development of a new family of devices for both classical and quantum computing. This review is intended to detail the work of the past twenty years concerning the link between spin-glass theory and optics. After a simple introduction to the main ideas of spin glasses, we start from the first works aimed at finding a direct experimental proof of ideas such as the landscape and ultrametricity; then we report on “linear optical spin glasses,” which refer to the photonic simulation of various Ising models for combinatorial optimization and interlinked with quantum computers; finally, we discuss the emerging field of “nonlinear optical spin glasses,” driven by the impressive progress in the realization of coherent Ising machines with parametric oscillators, that opened an new research direction driven by the cross-fertilization of advanced theoretical physics, artificial intelligence, classical and quantum nonlinear optics.

Ising Machine by Dimensional Collapse of Nonlinear Polarization Oscillators

https://journals.aps.org/prl/abstract/10.1103/qs29-2xqc

Phys. Rev. Lett. 135, 063801 – Published 4 August, 2025

Ising machines show promise as ultrafast hardware for optimizations encoded in Ising Hamiltonians but fall short in terms of success rate and performance scaling. Here, we propose a novel Ising machine that exploits the three-dimensional nature of nonlinear polarization oscillators to counteract these limitations. Based on the evolution of the optical polarization in third-order nonlinear media, the high-dimensional machine reaches the Ising ground state by the mechanism of “dimensional collapse”: the dynamics on the Poincaré sphere undergoes a self-induced collapse into polarization fixed points mapping an Ising spin. Collapse from a spherical to a binary spin occurs when the polarization oscillator experiences iterative loss and anisotropic feedback. The photonic setup consists of polarization modulated pulses in a 𝜒(3) crystal subject to measurement and feedback. We numerically demonstrate the polarization machine achieves enhanced success probability on benchmark graphs and an exponential improvement in performance scaling with respect to coherent Ising machines due to its high-dimensional operation. The proposed Ising machine paves the way for a new class of Ising solvers with enhanced computing capabilities.

Equalized Hyperspin Machine

The reliable simulation of spin models is of critical importance to tackle complex optimization problems that are intractable on conventional computing machines. The recently introduced hyperspin machine, which is a network of linearly and nonlinearly coupled parametric oscillators, provides a versatile simulator of general classical vector spin models in arbitrary dimension, finding the minimum of the simulated spin Hamiltonian and implementing novel annealing algorithms. In the hyperspin machine, oscillators evolve in time minimizing a cost function that must resemble the desired spin Hamiltonian in order for the system to reliably simulate the target spin model. This condition is met if the hyperspin amplitudes are equal in the steady state. Currently, no mechanism to enforce equal amplitudes exists. Here, we bridge this gap and introduce a method to simulate the hyperspin machine with equalized amplitudes in the steady state. We employ an additional network of oscillators (named equalizers) that connect to the hyperspin machine via an antisymmetric nonlinear coupling and equalize the hyperspin amplitudes. We demonstrate the performance of such an equalized hyperspin machine by large-scale numerical simulations up to 10000 hyperspins. Compared to the hyperspin machine without equalization, we find that the equalized hyperspin machine (i) Reaches orders of magnitude lower spin energy, and (ii) Its performance is significantly less sensitive to the system parameters. The equalized hyperspin machine offers a competitive spin Hamiltonian minimizer and opens the possibility to combine amplitude equalization with complex annealing protocols to further boost the performance of spin machines.

[2507.12940] Equalized Hyperspin Machine

Phys. Rev. A 112, 053505 (2025)

The First Experimental Observation of Ultrametricity

https://www.researchsquare.com/article/rs-5433512/v1

Ultrametricity is a fundamental mathematical concept that describes a particular metric space in which every triplet of points in the space forms an isosceles triangle. The ultrametric space differs from the usual Archimedean metric, where three points are allowed from any triangle.

Ultrametricity is the topology of hierarchical architectures. Examples can be found in taxonomy, where phylogenetic trees are ultrametric, mathematics with p-adic numbers, geography for measuring landscape complexity, and physics, where complex systems have intrinsically an ultrametric structure.

The Noble Prize Giorgio Parisi demonstrated this within the theory of spin glasses, where the overlap between spins exhibits ultrametricity, with the mathematical solution given by the full replica symmetry breaking.

An experimental demonstration of this is still lacking due to the difficulty of finding measurable physical observables.

In 2015, we introduced random lasers as photonic counterparts of spin glasses, and we demonstrated the replica symmetry breaking by directly measuring the overlap between spins, known as the order parameter in the description of glass phase transitions.

In the work, we clearly show the hierarchical organization of the overlap matrix reproducing the Parisi Ansatz, and we experimentally prove the ultrametric nature of the replica states.

For the first time, we measure the distance between any three replicas forming a triangle, and we report the growth of the distribution of isosceles tringles when the system enters the glassy regime. This is an unambiguous way to demonstrate ultrametricity and has been previously done only in numerical simulations.

In addition, from the hierarchical structure of the spin states, illustrated as dendrograms, and the distances between replicas, we attain the first topological energy landscape of a complex system from experiments.

The great potentiality of our research is the ability to access measurable spins from emission spectra and to quantify the overlap parameter. Random lasers are photonic spin glasses, as they manifest a clear phase transition from a paramagnetic ordered state to a glassy disordered one by increasing the system’s energy. Thanks to this powerful asset, we demonstrate the ultrametricity of the replica space. We report the experimental energy landscape with a topology that changes from a flat large basin to the coexistence of many metastable minima and the braking of ergodicity in the glassy state.

Emergent Equilibrium in All-Optical Single Quantum-Trajectory Ising Machines

We investigate the dynamics of multi-mode optical systems driven by two-photon processes and subject to non-local losses, incorporating quantum noise at the Gaussian level. Our findings show that the statistics from a single Gaussian quantum trajectory exhibit emergent thermal equilibrium governed by an Ising Hamiltonian encoded in the dissipative coupling between modes. The driving strength sets the system’s effective temperature relative to the oscillation threshold. Given the ultra-short time scales typical of all-optical devices, our study demonstrates that such multi-mode optical systems can operate as ultra-fast Boltzmann samplers, paving the way toward the realization of efficient hardware for combinatorial optimization, with promising applications in machine learning and beyond.

https://arxiv.org/abs/2412.12768

https://mathstodon.xyz/@nonlinearxwaves/113672283856089363