Analog spin simulators: How to keep the amplitude homogeneous

W. Verstraelen, P. Deuar, M. Matuszewski, and T.C.H. Liew

Phys. Rev. Applied 21, 024057 – Published 29 February 2024

Abstract

A setup that simulates ground states of spin graphs would allow one to solve computationally hard optimization problems efficiently. Current optical setups to this goal have difficulties decoupling the amplitude and phase degrees of freedom of each effective spin, which risks that the resultant mapping will be invalid, a problem known as amplitude heterogeneity. Here, we propose a setup with coupled active optical cavity modes, where this problem is eliminated through their particular geometric arrangement. Acting as an effective Monte Carlo solver, the ground state can be found exactly. By tuning a parameter, the setup solves $X Y$ or Ising problems.

Received 8 June 2023
Revised 7 December 2023
Accepted 5 February 2024

DOI:https://doi.org/10.1103/PhysRevApplied.21.024057

Physics Subject Headings (PhySH)

Analog computation Polaritons

Coupled oscillators Polariton condensate

Ising model Optical computing Phase space methods XY model

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

W. Verstraelen ^1,*, P. Deuar ², M. Matuszewski², and T.C.H. Liew¹

¹Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
²Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, Warsaw PL-02-668, Poland

^*wouter.verstraelen@ntu.edu.sg

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Vol. 21, Iss. 2 — February 2024

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Images

Figure 1
Depiction of the proposed scheme with polariton micropillars acting as coupled cavities. Two “spin” sites $ψ_{n}$ and $ψ_{m}$ are coherently coupled through the undriven “connecting” $χ_{n m}^{S}$ and $χ_{n m}^{A}$ . The bump in the coupling between $χ_{n m}^{A}$ and $ψ_{m}$ indicates its different path length giving this coupling a strength $- J$ , while the others are all $+ J$ . Inset: $ψ_{n, m}$ correspond to a generic pair of connected spins in the graph, with either ferromagnetic or antiferromagnetic coupling.
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Figure 2
Amplitude heterogeneity. Consider an Ising model with $H = - \sum_{i j} J_{i j} s_{i} s_{j}$ on a triangular graph with two ferromagnetic ( $J_{i j} = 1$ ) and one antiferromagnetic ( $J_{i j} = - 1$ ) coupling, depicted in green (red), for variables $s = \pm 1$ . It is frustrated: no Ising configuration can satisfy all couplings simultaneously. One of the degenerate ground states is shown. However, the mapping from spins $s_{i}$ to complex amplitudes $ψ_{i}$ is not exact, since the absolute values $| ψ_{i} |$ are not normalized to unity. This leads to lower-energy solutions, which are unphysical in terms of the original Ising model. In this work, a scheme is introduced that avoids the occurrence of such amplitude heterogeneity.
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Figure 3
Absolute amplitude (radial component) and phase (angles) for a random graph composed of seven “spin” sites (red), eight “extra” sites (green) that ensure the main sites have all the same number of neighbors (four for the considered example), and “connecting” sites (blue). Time evolution is shown with a linearly increasing opacity, up to a stationary state denoted by the spots. (a) $X Y$ -type behavior where the “spin” sites adopt arbitrary phases; (b) Ising-type behavior where the “spin” site phases become binary. The black circle corresponds to the analytically predicted magnitude $\sqrt{I_{0}}$ . Parameters: $J / Γ_{NL} = 0.5$ ; $P / Γ_{NL}, = 12 + 2$ ; $γ / Γ_{NL} = 4$ ; and $Γ_{NL}^{'} / Γ_{NL} = 0$ in the $X Y$ case (a) or Ising case (b). The graph has a 50% connectivity, all antiferromagnetic to ensure frustration to make it nontrivial. Even though the “extra” sites are used, we have added contribution +2 to $P / Γ_{NL}$ that compensates exactly for the losses $8 n_{NB} J^{2} / γ$ through the neighboring sites.
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Figure 4
Option 1 (all in one picture). Solving $X Y$ (a)–(d) and Ising (e)–(h) graphs. Panels (a)–(c)[(e)–(g)] depict the solution of the same random $X Y$ [Ising] graph and parameters (with $P_{r}$ taking the value of $P$ ) of (3). (a)[(e)] The spin energy stabilizes when the feedback strength $P_{f} \propto μ (j)$ becomes sufficiently strong. An example of a rising $μ (j)$ is schematically added. (b)[(f)] as analytically predicted, $E_{spin}$ is directly proportional to $⟨ | χ |^{2} ⟩$ , where $⟨ \cdot ⟩$ denotes the spatial average, selecting either $χ^{S}$ or $χ^{A}$ out of every coupling—allowing the scheme to work. (c)[(g)] Compares the distribution of $X Y$ [Ising] energies $E_{spin}$ from the photonic simulator across pulses $j$ (blue) with the raw density of states (as obtained from a brute-force search) (orange). These results show that the ground-state solution is exactly found, and as the simulator remains in this configuration, it forms a peak, which keeps on growing indefinitely. This behavior was verified for numerous individual graphs (d)[(e)] Statistics in the performance of different random graphs at multiple sizes, in finite time. Depicted is the excess energy above the true ground state $Δ E_{spin}$ as a function of a rescaled number of pulses (yellow-red for different graph sizes $N_{sites} = 2 - 12$ , each datapoint is the average over 30 graphs). Each curve shows that $Δ E_{spin}$ and hence the distance from the true ground state becomes vanishingly small for sufficient pulses. The collapse for different sizes demonstrates the scaling behavior that allows one to extrapolate that the scheme would apply to graphs of arbitrary size. For these graphs, 50% all-to-all connectivity was chosen, which in turn has 50% ferromagnetic or antiferromagnetic coupling, reflecting realistic large graph problems. To compensate for external losses, in the $X Y$ case (d) extra dangling sites were used in combination with offset in $P_{r}, P_{f}$ , while in the Ising case (h) only the offset was used without extra dangling sites. Results of the Ising case are less noisy since the spin states remain gapped (arbitrary perturbations would not lead to other configurations). All pulses used a total duration of $1000 Γ_{NL}^{- 1}$ .
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