Four-body interactions in Kerr parametric oscillator circuits

TL;DR

This work demonstrates that Kerr nonlinearities in Kerr parametric oscillators (KPOs) can realize intrinsic four-body interactions suitable for Lechner-Hauke-Zoller (LHZ) parity encoding, enabling scalable embedding of all-to-all Ising couplings without nonlinear couplers. The authors develop unit circuit designs, derive and compare four-body couplings from KPO nonlinearities ($g^{(4)}$, $h^{(4)}$, and $ ilde{h}^{(4)}$), and validate the concept experimentally with a four-KPO device showing four-body correlations and LHZ-like quantum annealing. Numerical results indicate that four-body couplings can reach magnitudes comparable to coupler-mediated schemes, and locality/scalability analyses show feasible tiling into larger networks with careful pump-frequency management. The study also discusses residual interactions and cross-Kerr cancellation strategies, outlining a practical path toward simpler, scalable Ising machines and broader quantum-information processing applications.

Abstract

We theoretically present new unit circuits of Kerr parametric oscillators (KPOs) with four-body interactions, which enable the scalable embedding of all-to-all connected logical Ising spins using the Lechner-Hauke-Zoller (LHZ) scheme. These unit circuits enable four-body interactions using linear couplers, making the circuit fabrication and characterization much simpler than those of conventional unit circuits with nonlinear couplers. Numerical calculations indicate that the magnitudes of the coupling constants can be comparable to those in conventional circuits. On the basis of this theory, we designed a four-KPO circuit and experimentally confirmed the four-body correlation by measuring the pump-phase dependence of the parity of the four-KPO states. We show that the choice of the pump frequencies are important not only to enable the four-body interaction, but to cancel the effects of other unwanted interactions. Using the circuit, we demonstrated the quantum annealing based on the LHZ scheme, where the strength of the interaction between the logical Ising spins is mapped to the local field and controlled by a coherent drive applied to each KPO.

Figures (14)

• Figure 1: Schematic of a conventional unit circuit with a four-body interaction between four KPOs. We label the KPOs as KPO 1 to KPO 4. A box with a cross mark inside represents a Josephson junction. The $\varphi_k$ ($k=1,...,6$) is the reduced node flux, $C_q$s and $C_g$ are the cavity capacitances of the KPOs and coupler, respectively, $C_c$s are the coupling capacitances, and $L_q$s and $L_g$ are the linear inductances of the KPOs and coupler, respectively.
• Figure 2: Unit circuits with four-body interactions originating from KPO nonlinearities. The black boxes represent inductive components of the KPOs. In (a), four KPOs are capacitively coupled to each other without going through any grounded elements. In (b), KPOs 1 and 2, 1 and 3, and 2 and 3 are coupled via KPO 4, resulting in the suppression of $h_{12}$, $h_{13}$, and $h_{23}$. In (c), KPOs 2 and 3 are coupled via KPOs 1 and 4, resulting in the suppression of $h_{23}$.
• Figure 3: Abstractions of capacitive components. The white circles, navy diamonds, and navy circles represent grounded KPOs, ungrounded couplers, and nodes (terminals of connections), respectively. The double lines indicate connecting wires interrupted by a coupling capacitance $C_c$. (a) Unit circuit with an ungrounded coupler. (b) Extension of (a). There are two ungrounded couplers and seven KPOs. (c) Unit circuit without a coupler.
• Figure 4: KPO arrangement based on the LHZ scheme Sourlas:2005Lechner:2015bal with $h^{(4)}$. The regions outlined by the red dashed squares correspond to a unit cell. In (a), there is one KPO (orange) that has a different Kerr nonlinearity from other three KPOs (blue) in each unit circuit. In (b), there are two KPOs that have different Kerr nonlinearities from other two KPOs in each unit circuit.
• Figure 5: Large-scale network based on the unit circuit in Fig. \ref{['fig:circuits_qubit_4bi']}(b), where the four-body interaction originates from only one Kerr nonlinearity ($K_4$ in the circuit). The subset of the green KPOs represents a unit cell as an example. Note that the region outlined by the orange dashed square also represents a unit cell. There is a four-body interaction between the red KPOs, which originates from the yellow KPO.