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arXiv:2401.08055

A Programmable and Reconfigurable Photonic Simulator for Classical XY Models

Jiayi Ouyang, Yuxuan Liao, Xue Feng, Yongzhuo Li, Kaiyu Cui, Liu, Fang, Hao Sun, Wei Zhang, Yidong Huang

Published Jan 16, 2024• 0 citations
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Abstract

In this work, we proposed and experimentally demonstrated a photonic simulator for XY models, which is a typical kind of classical spin models. By encoding the XY spins on the phase term of the input light field, the corresponding XY Hamiltonian could be performed on the output light intensities. The simulator is mainly based on a programmable and reconfigurable optical vector-matrix multiplication system, which can map arbitrary XY models within the dimensionality limit. Here, we demonstrated the Berezinskii-Kosterlitz-Thouless transition in a two-dimensional XY model, in which the expectation values of some observables are calculated and consistent with the theory. Besides, we performed the ground state search of two 25-spin XY models with different spin connections and coupling strengths. Our proposal paves a new way to investigate the XY spin system.

DOI
10.48550/arxiv.2401.08055

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In this work, we proposed and experimentally demonstrated a photonic simulator for XY models, which is a typical kind of classical spin models. By encoding the XY spins on the phase term of the input light field, the corresponding XY Hamiltonian could be performed on the output light intensities. The simulator is mainly based on a programmable and reconfigurable optical vector-matrix multiplication system, which can map arbitrary XY models within the dimensionality limit. Here, we demonstrated the Berezinskii-Kosterlitz-Thouless transition in a two-dimensional XY model, in which the expectation values of some observables are calculated and consistent with the theory. Besides, we performed the ground state search of two 25-spin XY models with different spin connections and coupling strengths. Our proposal paves a new way to investigate the XY spin system.
The ability to simulate XY models of classical spins is rather important, since e.g. it is related to solving NP-hard optimization problems. This study uses a photonic simulator to realize XY Hamiltonians with arbitrary spin connections and coupling strengths. The key unit is an optical system for vector-matrix multiplication that can perform arbitrary transformations of complex matrices. The Berezinskii-Kosterlitz-Thouless transition and ground-state search of several XY models are demonstrated experimentally. Thus this Letter provides an effective alternative approach for investigating such models and solving continuous quadratic optimization problems with optical systems.
The general-purpose programmable photonic processors offer a scalable and reconfigurable solution for a wide range of RF and optical applications. Therefore, implementing photonic Ising machines using programmable processors leverages the advantages of high speed and parallelism, enabling efficient hardware acceleration for finding ground-state solutions to combinatorial optimization problems. In this work, we demonstrate a novel programmable photonic Ising solver based on a hexagonal mesh general-purpose programmable photonic platform. The integrated system allows reconfigurable matrix multiplication and computes the Hamiltonian iteratively using an annealing algorithm that facilitates spin updates and effectively searches for the ground state. As a proof of concept, we experimentally solve two benchmark optimization problems, a fundamental three-node ferromagnetic coupling problem with external bias that demonstrates nontrivial spin interactions, and a four-node Max-Cut problem with arbitrary coupling matrices. Furthermore, to establish a large-scale capability, we emulated Ising problems with sizes up to N = 50, achieving success probabilities exceeding 80\%. Additionally, we examined the impact of errors, such as phase and coupling, on the performance of the programmable photonic Ising machine. Our general-purpose photonic Ising machine paves the way for implementing large-scale, programmable architectures for solving optimization problems.
The ability to simulate physical systems is essential for an understanding of their physical properties and advancing related technologies. However, simulating certain complex systems at scale, including many-body spin models, is often a significant challenge for conventional computing approaches. Quantum simulators, employing controlled quantum systems that mimic the dynamics of a target system, can offer a more efficient approach. This work demonstrates the use of photonic circuits for quantum simulation of disordered spin systems, where quantum properties such as superposition and interference can be leveraged for a more efficient simulation. We fabricate and characterize a low-loss silicon-on-insulator chip with a buried aluminum mirror and a reconfigurable interferometer, which is employed in conjunction with two input photons to map the dynamics of a spin Hamiltonian with four-body interactions. Our results demonstrate several characteristics associated with the target system, such as spin-degenerate ground states and phase transitions, confirming the device’s functionality as a quantum simulator of the spin system.
We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.
It is argued that two-dimensional U(N) spin models for any N undergo a Berezinskii-Kosterlitz-Thouless (BKT)-like phase transition, similarly to the famous XY model. This conclusion follows from the Berezinskii-like calculation of the two-point correlation function in U(N) models, approximate renormalization group analysis, and numerical investigations of the U(2) model. It is shown, via Monte Carlo simulations, that the universality class of the U(2) model coincides with that of the XY model. Moreover, preliminary numerical results point out that two-dimensional SU(N) spin models with the fundamental and adjoint terms and N>4 exhibit two phase transitions of BKT type, similarly to Z(N) vector models.