Quantum annealing in capacitively coupled Kerr parametric oscillators using frequency-chirped drives

TL;DR

Quantum annealing with Kerr parametric oscillators: The paper demonstrates reliable preparation of the Ising-solution state using a pair of capacitively coupled KPOs under frequency-chirped drives. The method encodes the problem in an Ising energy $E_{Ising} = -J_{LR} s_L s_R + h_L s_L + h_R s_R$ with $J_{LR} = 2 \cos(\theta_p/2) \alpha_L \alpha_R g$ and local fields $h_j = 2 \sin(\theta_{s j}) \alpha_j \Omega_{d j}$, while detuning is dynamically swept by chirping the drives. The main findings show that frequency chirping increases the success probability for the solution state by reducing population transfer to excited states, improves phase locking (locking error <1% at $P_s = -105$ dBm) and boosts correlation between oscillators up to ~97%, in agreement with Lindblad master equation simulations including pure dephasing. The contributions demonstrate practical KPO-based quantum annealing and outline steps toward scalable KPO networks with engineered four-body embeddings and optimized drive profiles.

Abstract

We study parametric oscillations of two capacitively coupled Kerr parametric oscillators (KPOs) with frequency-chirped two- and one-photon drives. The two-KPO system adiabatically evolves from the initial vacuum state to an oscillation state corresponding to a solution state in quantum-annealing applications. Frequency chirping dynamically changes the detuning between resonance and oscillation frequencies during parametric modulation and reduces unwanted population transfer to excited states caused by pure dephasing and photon loss. We observe that frequency chirping increases the success probability to obtain the solution state and that simulations taking into account pure dephasing reproduce experiments with and without frequency chirping. This study demonstrates the effectiveness and applicability of frequency chirping to a KPO-based quantum annealer.

Figures (11)

• Figure 1: Schematic energy level diagram of KPO-based quantum annealing. We assume negative Kerr nonlinearity. Solid and dashed lines show eigenenergies of states which adiabatically evolve from vacuum and non-vacuum states, respectively, as a function of the amplitude of parametric modulation. Black, blue and red lines show eigenenergies with frequency chirping, zero detuning, and a fixed detuning $\delta<0$, respectively. Insets show schematic one-bit Wigner functions at the start and the end of the parametric modulation.
• Figure 2: Schematic circuit diagram of device chip. Grey rectangular region represents chip. Each JPO labeled L and R consists of symmetric DC-SQUID (superconducting quantum interference device) (shown in red) embedded in center of transmission-line resonator (grey tubes). Each DC-SQUID is inductively coupled to on-chip pump line. Coupling capacitor $C_{\rm c}$ and input capacitors $C_{\rm in}$ are shown in green and blue, respectively. Four ports of chip are labeled as LI, LP, RI, and RP. Circulators outside LI and RI ports route output microwaves.
• Figure 3: Phase locking of JPO (KPO) L. (a) Pulse sequence of two-photon drive, one-photon drive, and output of parametric oscillations. Cyan and purple represent frequencies of pulses $\omega_{\rm p}/2$ and $\omega_{\rm p}$, respectively. Shaded and hatched regions represent frequency chirping and integration time of heterodyne measurement, respectively (see Appendix \ref{['sup:pulse_parameter']} for details). (b) $IQ$-plane histogram of output with frequency chirping. Relative phases of $IQ$ amplitudes are subtracted analytically to align two peaks on $Q$ axis. Magnitudes of $IQ$ amplitudes are normalized by standard deviation of $I$ amplitudes. (c) $P_{\rm s \hbox{L}}$ and $\theta_{\rm s \hbox{L}}$ dependence of occurrence probabilities of $\ket{+\alpha_{\rm \hbox{L}}}$ with frequency chirping. (d) $P_{\rm s \hbox{L}}$ dependence of locking error. Locking errors are deduced by subtracting maximum observed probability of $\ket{+\alpha_{\rm \hbox{L}}}$ at each $P_{\rm s\hbox{L}}$ from unity. Black data shows locking error with frequency chirping, which is deduced from data shown in (c). Red and green data show locking errors deduced from experiments (not shown) using same experimental parameters as (c) without frequency chirping at $\Delta=0$ and $\Delta/2\pi=-20$ MHz, respectively.
• Figure 4: Simulation of phase locking. (a) Simulated pure-dephasing-rate dependence of locking error with frequency chirping. Simulation parameters are fixed to experimental ones used for data with frequency chirping and highest signal power shown in Fig. \ref{['fig:phaselock']}(d), where we fix $\theta_{\rm s\hbox{L}}=3\pi/2$. Dashed red line with band represents corresponding experimental locking error with statistical uncertainty (standard deviation). (b) Simulated signal-power dependence of locking error at $\gamma_{\rm \hbox{L}}/2\pi=6.8$ kHz. Markers and solid lines show experimental data shown in Fig. \ref{['fig:phaselock']}(d) and simulated data, respectively. Simulation parameters are fixed to experimental ones used for data in Fig. \ref{['fig:phaselock']}(c), and simulated locking errors are deduced in same manner as in Fig. \ref{['fig:phaselock']}(d). Black circles show locking error with frequency chirping. Red crosses and green triangles show locking errors obtained without frequency chirping at $\Delta=0$ and $\Delta/2\pi=-20$ MHz, respectively.
• Figure 5: Simultaneous parametric oscillations without one-photon drives. Trapezoidal two-photon drives are simultaneously applied to KPOs, then outputs are measured (see Appendix \ref{['sup:pulse_parameter']} for details). (a) Histogram of $Q$ amplitudes of outputs with frequency chirping. $Q$ amplitude of each KPO is normalized by standard deviation of $I$ amplitude. (b) Pump phase $\theta_{\rm p}$ dependence of same-phase probability. Same-phase probability is sum of observed probabilities of same-phase states $\ket{\pm \alpha_{\rm \hbox{L}}}\ket{\pm\alpha_{\rm \hbox{R}}}$. Black circles (1) show experimental data obtained with frequency chirping. Red crosses and green triangles (2, 3) show experimental data obtained without frequency chirping at $\Delta=0$ and $\Delta/2\pi=-20$ MHz, respectively. Solid lines show simulated dependence with and without frequency chirping, where we assume $\gamma_{\rm \hbox{L(R)}}/2\pi=7.7$ kHz. (c) Detuning dependence of same-phase probability with frequency chirping. Horizontal dashed line shows coupling strength $-g$.