Spin Model for Quantum Annealing with Kerr Parametric Oscillators

Abstract

Coherent states offer a promising path for near-term quantum computing due to their inherent protection against bit-flip noise. However, their large photon numbers can be challenging for numerical simulation. This paper introduces an effective model, representing coherent-state quantum annealing using spin-1/2 degrees of freedom. We demonstrate that this model yields accurate predictions for realistic experimental settings and can therefore serve as a practical tool for optimizing future quantum hardware.

Figures (4)

• Figure 1: Most likely annealing outcomes as function of single-photon drive strengths $\tilde{\Omega}_{0,1}$, and single-photon drive schedules, parameterized by $f$. Data are computed using the spin model. The colors on the heat map represent the most likely final states. The black lines indicate the ground-state boundaries of the final Hamiltonian, for all values of $f$. The simulations suggest that the outcomes deviate from ideal state boundaries, indicating that our assumption of adiabaticity, used in \ref{['eq:method:alpha_max']}, can be violated.
• Figure 2: Comparison of Fock and spin simulation for two KPOs. Spin simulations correspond to data shown in \ref{['fig:2kpo:diagram']} for fixed $\tilde{\Omega}_0/2\pi = \qty{9}{\mega\hertz}$. Markers indicate Fock simulations, while the solid lines are results of the spin model. Probabilities for state $\left| -+\right\rangle$ are very close to zero for all methods and parameters, and are thus not shown. The black, vertical line indicates the single-photon drive strength $\tilde{\Omega}_1$ where $\left| ++\right\rangle$ and $\left| +-\right\rangle$ are degenerate in energy. The spin model tends to overestimate the probability of the most likely spin configuration. However, the spin model can quite accurately predict the transition point, i.e., the value of $\tilde{\Omega}_1$ where the most-likely state changes. For $f\geq 1.68$ we observe cusps in the spin results, which appear not to be present in the Fock-space simulations.
• Figure 3: Outcome probabilities for the ideal ground states as functions of the local single-photon drive on KPO $3$. The most likely state changes from $\left| +-+-\right\rangle$ to $\left| +-++\right\rangle$ for larger $\tilde{\Omega}_3$. Markers correspond to simulations in the Fock space, while the solid lines are the result of simulations in the spin model. Compared to \ref{['fig:2kpo:spin_fock']}, the schedule of the local fields, parameterized by $f$, has a much smaller influence on the observed results.
• Figure 4: Illustration of the schedules used in our simulations. The parameters are taken directly from the recent experiment Yamaji2025. The state is prepared during $T_\mathrm{s}$, then the single-photon drive $\Omega(t)$ needs to be turned off again, before the KPOs are measured during time $T_\mathrm{r}$, indicated by the shaded region. We use a monomial schedule with exponent $2.5$ for the two-photon drive $P(t)$, as in the experiment Yamaji2025, but vary the exponent $f$ of the local field-schedule $\Omega(t)$.