Equalized Hyperspin Machine

TL;DR

The paper addresses amplitude heterogeneity in hyperspin machines that map networks of degenerate parametric oscillators to $D$-vector spin Hamiltonians. It introduces an equalization layer of auxiliary oscillators with antisymmetric nonlinear coupling to enforce equal hyperspin amplitudes, yielding an equalized steady state where the hyperspin energy more closely tracks the $D$-vector Hamiltonian $H_D$; the steady-state amplitude satisfies $S^2=rac{1}{eta}igl(rac{h}{2}-g-rac{H_D}{N}igr)$ and the threshold $h_{ m min}=2igl(g+E_{D, m min}/Nigr)$, with finite-size scaling $J o sJ$ required to maintain $S>0$ as $N$ grows. Large-scale simulations up to $N=10^4$ hyperspins show that equalization dramatically reduces the minimum energy reached and suppresses amplitude heterogeneity, while remaining robust to parameter variations and compatible with dimensional annealing approaches. These results establish equalized hyperspins as a competitive, more reliable spin-Hamiltonian minimizer, expanding the practical utility of oscillator-based spin machines for complex optimization tasks.

Abstract

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.

Figures (10)

• Figure 1: Hyperspin fixed points from Eq. \ref{['eq:nonlineardynamicsparametricoscillator02']} with different $D$. (a) Ising spin $D=1$, (b) XY spin $D=2$, and (c) Heisenberg spin $D=3$. The fixed points are shown in the $(A_1,A_2,A_3)\subseteq\mathbb{R}^3$ space. For the Ising spin there is one oscillator only so $A_2=A_3=0$ in the figure. Similarly for the XY spin there are two oscillators, so $A_3=0$. The red dots are the fixed points of the dynamics obtained by integrating Eq. \ref{['eq:nonlineardynamicsparametricoscillator02']} several times for different initial conditions. The fixed points are found on the boundary of a $D$-dimensional surface (hypersphere), which in particular is a segment, a circle, and a sphere for $D=1,2,3$, respectively. The oscillator amplitudes can then be used to define a $D$-dimensional vector (blue arrow), which simulates a $D$-dimensional spin (hyperspin). Bottom panels depict the nonlinearly coupled oscillators (left panels) and the corresponding representation as hyperspin vectors $\vec{\sigma}_1=\vec{S}_1/S_1$ (right panels), for the same $D$ as in the corresponding top panel. Colored dots depict the individual oscillator amplitudes, subject to intrinsic loss (quantified by $g$, green wavy arrows), and the outer gray circle with gray arrow depicts the pump ($h$) that is saturated by all the oscillators at once. The geometrical interpretation is here shown for $D=1,2,3$ for illustration purposes, but it is valid for any $D$ (see Ref. strinati2022hyperspinmachine for an extended discussion).
• Figure 2: (a) Scheme of the linear coupling matrix $\mathbf{C}$ between any two hyperspins $\vec{S}_q$ and $\vec{S}_p$ with generic $\mathbf{G}_D$ [Eqs. \ref{['eq:nonlineardynamicsparametricoscillator08']} and \ref{['eq:nonlineardynamicsparametricoscillator09']}], with $D=3$ for illustration purposes. Hyperspins are represented as in Fig. \ref{['fig:sketchfixedpoints02']} and the linear coupling connections are depicted by the black lines between the colored dots. (b) As in panel (a) but with $\mathbf{G}_D=\mathbbb{1}_D$, which implements the standard Euclidian metric. Graphically, only the connections between dots of the same color (which are the homologous vector components between the two hyperspins) are nonzero. (c) Representation of the hyperspin machine used throughout this paper, obtained by connecting hyperspins with arbitrary $\mathbf{J}_N$. (d) Scheme of the working principle of the hyperspin machine with unconstrained hyperspin amplitudes. The $D$-vector model energy $H_D$ [blue line, from Eq. \ref{['eq:dynamicsamplitueheterogeneity07']}] computed from the hyperspin amplitudes in time decreases and reaches a steady-state value that can be significantly larger than the actual global minimum $E_{D,{\rm min}}$ [red dashed line, from Eq. \ref{['eq:amplitudehomogeneityhyperpsin18bis7bis1']}].
• Figure 3: (a) Connectivity of two hyperspins formed by parametric oscillators $A_{j}$ (colored dots in the gray circles) to two auxiliary oscillators $Y_{k}$ named equalizers (yellow dots) performing amplitude equalization. Here $N=2$ and $D=2$ (XY hyperspins) are chosen for illustration purposes. The oscillator amplitudes $A_1$, $A_2$, $A_3$, and $A_4$ define two XY vectors $\vec{S}_1=(A_1,A_2)$ and $\vec{S}_2=(A_3,A_4)$, coupled by a linear coupling implementing the Euclidian scalar (dot) product $\vec{S}_1\cdot\vec{S}_2$ (black lines). The PO amplitudes $A_j$ are coupled by a directional, antisymmetric nonlinear coupling (see Table \ref{['table:couplingandtypeofcouplinghyperspinmachine01']}) with coupling constant $\delta_{\rm eq}>0$ to the two equalizers $Y_1$ and $Y_2$, whose role is to enforce $S_1=S_2\equiv S$ in the steady state (see Sec. \ref{['sec:examplecorrectionintwohyperspins01']}). (b) Illustration of the directionality of the coupling between oscillators and equalizers. Solid blue (dashed magenta) line denotes that the coupling constant $\delta_{\rm eq}$ appears with the minus (plus) sign when going from the POs $A_j$ to the equalizers $Y_k$, and with plus (minus) sign when going from the equalizers to the POs [Eqs. \ref{['eq:dynamicsamplitueheterogeneity0101']}-\ref{['eq:dynamicsamplitueheterogeneity0103']}].
• Figure 4: Scheme of the multi-layer connectivity of the hyperspin machine with amplitude heterogeneity correction, for generic $N$ and $D$ (here $D=3$ is chosen for illustration purposes only). The $N$, $D$-dimensional hyperspins $\vec{S}_q$ are ordered on an open chain. Each pair of hyperspins $\vec{S}_q$ and $\vec{S}_{q+1}$ couples to a pair of equalizers as explained in Fig. \ref{['fig:schemeamplitudeheterogeneity1']}, which enforce $S_q=S_{q+1}\equiv S$ (pair amplitude equalization) in the steady state. Pair amplitude equalization is enforced by the full layer of equalizers for all $q$, therefore achieving the conditions $S_q\equiv S$ for all $q$ (amplitude equalization of all hyperspin amplitudes). The three couplings (nonlinear symmetric $\mathbf{W}$ between POs forming an hyperspin, linear symmetric $\mathbf{C}$ between POs in different hyperspins, and nonlinear antisymmetric $\mathbf{V}$ between POs and equalizers) are also shown.
• Figure 5: Finite size scaling of (a) Average minimum energy $E_{D,{\rm min}}$ [Eq. \ref{['eq:amplitudehomogeneityhyperpsin18bis7bis1']}] in absolute value, computed by python numerical minimizer (see text), (b) Largest eigenvalue $\lambda_{\rm max}$ [Eq. \ref{['eq:amplitudehomogeneityhyperpsin18bis7bis2']}], and (c) Squared equalization amplitude $S^2$ [Eq. \ref{['eq:amplitudehomogeneityhyperpsin18bis8']}] for $\delta h=0.4$, $\beta=0.1$, and $g=1$, where the black horizontal line marks the value $S^2=0$ below which no equalized solution exists. Averages are computed over $100$ random sparse binary graphs $\mathbf{J}$, for $D=2$ and $N$ ranging from $10$ to $500$. The uncertainties (shaded areas) are quantified by the interquartile range (IQR). The graph average is denoted by $\langle\cdot\rangle$. The data in panels (a) and (b) are fitted by the power law $f_l(N)=c_l\,N^{d_l}$ where $l=E,\lambda$, respectively (green dashed line), with fitted parameters $(c_E,d_E)\simeq(0.093,1.525)$ for the energy scaling and $(c_\lambda,d_\lambda)\simeq(0.104,0.530)$ for the eigenvalue scaling (note that $d_E-d_\lambda\simeq1$). One sees from panel (c) that $S^2$ decreases with $N$ and becomes negative already for $N\gtrsim40$, implying that the equalization point ceases to exist as $N$ increases (red area). (d) Rescaled energy $E_{D,{\rm min}}/N^\alpha$ and (e) Maximum eigenvalue $\lambda_{\rm max}/N^\alpha$ for $\alpha=d_E-1\simeq0.525$, obtained by multiplying $\mathbf{J}$ by $1/N^\alpha$ for all $N$. This scaling makes the energy increase linearly with $N$, highlighted in the panel (d) by the green dashed line, and the maximum eigenvalue in panel (e) becomes almost independent of $N$. Accordingly, $S^2/N^\alpha$ in panel (f) becomes fairly constant in $N$, ensuring the existence of the equalization point for all $N$. Note the log-log scale in panels (a),(b), and (d), and the semi-log scale in panels (c), (e), and (f).