Three Strongly Coupled Kerr Parametric Oscillators Forming a Boltzmann Machine

Abstract

Coupled Kerr parametric oscillators (KPOs) are a promising resource for classical and quantum analog computation, for example to find the ground state of Ising Hamiltonians. Yet, the state space of strongly coupled KPO networks is very involved. As such, their phase diagram sometimes features either too few or too many states, including some that cannot be mapped to Ising spin configurations. This complexity makes it challenging to find and meet the conditions under which an analog optimization algorithm can be successful. Here, we demonstrate how to use three strongly coupled KPOs as a simulator for an Ising Hamiltonian, and estimate its ground state using a Boltzmann sampling measurement. While fully classical, our work is directly relevant for quantum systems operating on coherent states.

Figures (5)

• Figure 1: Ising network built from three coupled KPOs. (a) The parametric phase states are represented as double-well quasipotentials whose minima are our artificial Ising spins. (b) Single KPO realized as a RLC circuit. We use a varactor diode as nonlinear capacitance $C$ and a coil as inductance $L$. The capacitor $C_\mathrm{b}$ and resistor $R_\mathrm{b}$ decouple coil and bias source. We use the voltage $U = U_\mathrm{F}\cos(\omega t) + U_\mathrm{p}\cos(2\omega t)$ to drive the resonator, while we measure the voltage signal $x$.
• Figure 2: Frequency sweeps. (a) Response of the three resonators to a frequency sweep with $U_\mathrm{p} = 250mV$ and $U_\mathrm{F} = 0$. Only the $u$ (in-phase) quadrature is shown. The signals of resonator 1 and 2 have been shifted by an offset $\pm\delta_\mathrm{u}$ for better visibility. (b) Phase diagram of the coupled system, measured as frequency up-sweeps at different $U_\mathrm{p}$. Shade qualitatively shows the response amplitude from low (bright) to high (dark). Colors indicate the configuration. Blue (around I): all three resonators move in phase with roughly equal amplitude. Yellow (around II): only two resonators have a non-zero amplitude and oscillate with roughly opposite phases. Magenta (around III): two resonators have the same phase, while the third one oscillates with opposite phase. White (remaining space): all resonators have zero amplitude. Labeled points mark the positions used in Fig. \ref{['fig:fig3']}. (c) Number of stable stationary solutions of Eqs. \ref{['eq:EOM']} calculated by Harmonic Balance kovsata2022harmonicbalance, encoded in the brightness contrast.
• Figure 3: Oscillation states measured at different positions in Fig. \ref{['fig:fig2']}(b). An external force at the measurement frequency $\omega$ is applied to each resonator before the parametric pump is switched on. By varying the force phases, different initial conditions are generated, triggering the system to choose different stationary oscillation solutions. The external forces are switched off when the parametric pump is switched on.
• Figure 4: Stochastic sampling of the network. The resonator system is driven parametrically at point III in Fig. \ref{['fig:fig2']}(b). Additionally, a white noise signal with a standard deviation of $137.88mV$ low-pass filtered at 5MHz is applied to the drive of each resonator, triggering switches between different phase states. The resonator's response is measured for $t_\mathrm{m} = 400s$ with a sampling rate of $\mathit{df} = 20.081kHz$. (a) 3D representation of the $u$ quadratures of a full dataset. Color encodes the configuration of the state as in Fig. \ref{['fig:fig3']}. Corners correspond to stable states, and jumps between them occur predominantly along the edges of the cube. The difference between the outermost blue and pink points appears exaggerated (relative to Fig. \ref{['fig:fig3']} III) due to the presence of noise and the difference in state occupation probability $P$. (b) Ising ground state prediction algorithm. Upper row shows the measured occupation probability; cf. data in (a) and supmat for details. The left column (i) is for identical coupling, the middle column (ii) is for $J_{12} < J_{31} < J_{23}$, and the right column (iii) has the same coupling as in the middle with an additional external force of $U_\textrm{F} = 5mV$ to break the symmetry between the phase states of each KPO. The middle and lower rows show the quasienergies $E_\textrm{p}$ of the KPO network in the rotating frame [cf. Eq. \ref{['eq:quasi_energy']}] and the eigenenergies $E_\textrm{I}$ for analogous configurations in a three-spin Ising Hamiltonian [cf. Eq. \ref{['eq:Ising']}].
• Figure 5: Inverting the nonlinearity. (a) Schematic of the quasipotential for a single parametric oscillator with positive $\beta$. (b) Numerical analysis of occupation probabilities $P$ for three resonators with $\omega_0 = 1$, $\lambda = 0.6$, $Q = 10$ and $\beta = 1$ and coupling $J_{12} = -1.5$, $J_{23} = -0.5$ and $J_{31} = -1$. Combined results for 30 numerical simulations of $t_\textrm{m} = 27.7h$ each, with a sampling rate of $df = 5Hz$. (c) Same as (a) for $\beta = -1$, and (d) same as (c) for negative $\beta$.