Complex Vector Gain-Based Annealer for Minimizing XY Hamiltonians

TL;DR

This paper presents the Complex Vector Gain-Based Annealer (CoVeGA), an analog computing platform designed to overcome energy barriers in XY Hamiltonians through a higher-dimensional representation, and introduces several graph structures that pose challenges for XY minimization and uses them to benchmark CoVeGA against single-dimension XY solvers.

Abstract

This paper presents the Complex Vector Gain-Based Annealer (CoVeGA), an analog computing platform designed to overcome energy barriers in XY Hamiltonians through a higher-dimensional representation. Traditional gain-based solvers utilizing optical or photonic hardware typically represent each XY spin with a single complex field. These solvers often struggle with large energy barriers in complex landscapes, leading to relaxation into excited states. CoVeGA addresses these limitations by employing two complex fields to represent each XY spin and dynamically evolving the energy landscape through time-dependent annealing. Operating in a higher-dimensional space, CoVeGA bridges energy barriers in this expanded space during the continuous phase evolution, thus avoiding entrapment in local minima. We introduce several graph structures that pose challenges for XY minimization and use them to benchmark CoVeGA against single-dimension XY solvers, highlighting the benefits of higher-dimensional operation.

Figures (8)

• Figure 1: (a) Ground state solution, up to a global phase change, for an $N = 12$ 4-regular Möbius ladder graph. In this ground state, every XY spin has a phase difference of $\pm 2 \pi / 3$ with each of its nearest neighbors. (b) Excited state of an $N = 102$ 4-regular Möbius ladder graph with $D = 4$. Here, the phases are not illustrated, and instead color represents the chilarity of each triangular base. For a base with indices $\{ i, j, k\}$, then $C_{ijk} = - 1$ is shown in yellow, $C_{ijk} = 1$ as red, and $C_{ijk} = 0$ as green. (c) Frequency density histogram of excitation parameter $D$ for $1000$ runs of the Kuramoto model on $N = \{12, 102, 204 \}$ 4-regular Möbius ladder graphs. In each run, initial phases $\theta_{i} (0)$ are chosen uniformly at random from range $[ - \pi, \pi )$, and Eq. (\ref{['Kuramoto Equation']}) is solved using the Euler scheme with fixed time step $\Delta t = 0.1$.
• Figure 2: (a) A ground state solution for an $N = 4 \times 4$ triangular lattice graph, where every XY spin has a phase difference of $\pm 2 \pi / 3$ with each of its neighbors. (b) Excited state of an $N = 10 \times 10$ triangular lattice graph. Here, the phases are not illustrated, and instead color represents the chirality of each triangular base. For a base with indices $\{ i, j, k \}$, then $C_{ijk} = - 1$ is show in yellow, $C_{ijk} = + 1$ as red, and $C_{ijk} = 0$ as green. Domains are more likely to exist when part of their boundaries are the edges of the triangular lattice since, on these edges, there are no excited (green) triangular bases. This is a consequence of the graph's non-periodic boundary conditions. (c) Frequency density histogram of excitation parameter $D$ for $1000$ runs of the Kuramoto model on $N = \{16, 100, 196 \}$ triangular lattice graphs. In each run, initial phases $\theta_i (0)$ are chosen uniformly at random from range $[-\pi, \pi)$, and Eq. (\ref{['Kuramoto Equation']}) is solved using the Euler scheme with fixed time step $\Delta t = 0.1$.
• Figure 3: (a) A configuration of XY phases on an $N = 4 \times 4$ basic Kuratowskian graph with $p = 0.2$. Ferromagnetic, antiferromagnetic, and zero couplings are illustrated as red, blue, and black lines, respectively. (b) Rank of coupling matrix $\mathbf{J}$ as a function of the probability parameter $p$. Each error bar in (b) corresponding to a different value of $p$ is constructed from 100 Kuratowskian graphs, each with random coupling weights generated by Eq. (\ref{['Kurat pmf']}). (c) Box plot distributions of sample variance values $s^2$ from final state energies of Eq. (\ref{['Kuramoto Equation']}), obtained from $N = 8 \times 8$ basic Kuratowskian graphs. Each box plot in (c) is obtained by sampling the final state energies over 100 runs on each of the 100 Kuratowskian graphs generated for each value of $p$.
• Figure 4: Panels (a) and (b) compare the trajectories of XY spins in the CoVeGA model and the single-dimension Stuart-Landau model, respectively, for an $N = 36$ 4-regular Möbius ladder graph. Red circles indicate initial states, while in (a), blue and cyan circles correspond to final states for $\psi_{i}^{(1)}$ and $\psi_{i}^{(2)}$. Final states $\psi_{i}$ in the scalar version (b) are highlighted with blue circles. Panels (c) and (d) illustrate the effective gain $\gamma_{i}$ as the systems evolve. CoVeGA successfully recovers the ground state without frustrations $D = 0$ as seen in inset (e), while the one-dimensional scalar version reaches the excited state $D = 2$ shown in inset (f). Both systems start from equivalent initial conditions, with the one-dimensional scalar version beginning at $\psi_{i} (0) = 0.01 \exp (i a)$ and CoVeGA at ${\bf \Psi}_{i} (0) = 0.01 ( \exp (i a), 0.1 \exp (i b))$, where $a$ and $b$ are uniformly chosen from range $[-\pi, \pi)$. Panel (g) compares the CoVeGA Hamiltonian $H$ against the one-dimensional model Eq. (\ref{['Hamiltonian Eq']}), while panel (h) shows the values of the XY Hamiltonians $H_{\rm XY}$.
• Figure 5: The probability distribution of recovering a state with excitation parameter $D$ for CoVeGA, one-dimensional Stuart-Landau, SVL, Kuramoto, and BFGS methods on $N = 12$ (blue), $102$ (orange), and $204$ (green) 4-regular Möbius ladder graphs. One thousand runs are used to calculate the probability of recovering excitation parameters $D$ for each value of $N$. Parameter values for CoVeGA are taken from Table (\ref{['Optimal_Params']}) in Appendix \ref{['Section:Materials']}wigley2016fast, and so are values for the Stuart-Landau network solver where applicable. For SVL, $m = 1.0$, $\xi \sim \mathcal{N} (0, 0.1)$, and $\gamma = 0.9$. In all cases, a fixed time step of $\Delta t = 0.1$ is used with annealing time length $T = 1000$.