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arXiv:2501.04981 · Japanese Journal of Applied Physics

Theoretical study of the spectroscopic measurements of Kerr nonlinear resonators with four-body interaction

Yuichiro Matsuzaki, Yuichiro Mori, Aiko Yamaguchi, Yohei Kawakami, Tsuyoshi Yamamoto

Published Apr 10, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract Quantum annealing provides a promising way to solve combinational optimization problems where the solutions correspond to the ground state of the Ising Hamiltonian. We can implement quantum annealing using the Kerr nonlinear resonators (KNRs), with bifurcation phenomena emerging when subjected to a parametric drive. These bifurcated states can function as bases of qubits. Moreover, integrating four-body interactions between physical qubits enables the establishment of effective all-to-all long-range interactions between logical qubits, which is essential for practical quantum annealing. While theoretical proposals exist for creating four-body interactions within KNRs, there has not been experimental verification through their spectroscopic signatures. In this paper, we theoretically investigate the spectroscopic measurements of KNRs featuring four-body interaction. We identify six distinct frequencies exhibiting population changes by employing resonant driving on one resonator and weak driving on another. Analytical and numerical calculations validate these findings. Our study demonstrates the potential of spectroscopy in characterizing systems with four-body interactions, offering insights for realizing quantum annealing with Kerr parametric oscillators.

DOI
10.35848/1347-4065/adcb7a

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Abstract Quantum annealing provides a promising way to solve combinational optimization problems where the solutions correspond to the ground state of the Ising Hamiltonian. We can implement quantum annealing using the Kerr nonlinear resonators (KNRs), with bifurcation phenomena emerging when subjected to a parametric drive. These bifurcated states can function as bases of qubits. Moreover, integrating four-body interactions between physical qubits enables the establishment of effective all-to-all long-range interactions between logical qubits, which is essential for practical quantum annealing. While theoretical proposals exist for creating four-body interactions within KNRs, there has not been experimental verification through their spectroscopic signatures. In this paper, we theoretically investigate the spectroscopic measurements of KNRs featuring four-body interaction. We identify six distinct frequencies exhibiting population changes by employing resonant driving on one resonator and weak driving on another. Analytical and numerical calculations validate these findings. Our study demonstrates the potential of spectroscopy in characterizing systems with four-body interactions, offering insights for realizing quantum annealing with Kerr parametric oscillators.
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.
Quantum annealing (QA) is an algorithm to find the ground state of the problem Hamiltonian by using an adiabatic time evolution. An approach to evaluate adiabaticity in the experiment by applying spectroscopic techniques has recently been suggested. However, this method requires temporal oscillation of interaction strength during QA, posing challenges for experimental demonstration. Here, we propose an experimental method for evaluating adiabaticity when performing QA with a parametric oscillator with Kerr nonlinearity (KPO). Importantly, our proposal offers a significant advantage by eliminating the need for temporal oscillation of interactions during QA. We investigate its performance through numerical simulations, and we show the feasibility of our method.
Abstract Quantum annealing (QA) is a way to solve combinational optimization problems. Kerr nonlinear parametric oscillators (KPOs) are promising devices for implementing QA. When we solve the combinational optimization problems using KPOs, it is necessary to precisely control the photon number of the KPOs. Here, we propose a feasible method to estimate the photon number of the KPO. We consider coupling an ancillary qubit to the KPO and show that spectroscopic measurements on the ancillary qubit provide information on the photon number of the KPO.
The Kerr parametric oscillator (KPO) has recently attracted considerable attention from the perspective of its applications to quantum information processing, and understanding its properties is an important challenge. Spectroscopic measurements serve as an effective means of elucidating detailed information about the system, such as the energy-level structure and the transition matrix elements of the KPO. Conventional spectroscopy requires the drive frequency to match an energy spacing with a nonzero transition matrix element. In recent years, a phenomenon called photonic resonance (PR) has been theoretically predicted in KPO spectroscopy. Specifically, resonance occurs under the condition that the detuning is set to n/2n/2n/2 times the Kerr nonlinearity, where nnn is a natural number. However, under this condition the transition matrix element vanishes, and thus the mechanism by which photonic resonance (PR) arises has remained unclear. In this work, we aim to elucidate the physical origin of PR observed in KPO spectroscopy. We first performed theoretical calculations and experiments of spectroscopic measurements, confirming that PR can indeed be observed and that the theoretical and experimental results are in qualitative agreement. We then carried out an analytical study under the assumption of an ideal noiseless environment. Our analysis revealed that, although the transition matrix element of the external field expressed in the system's energy eigenbasis is zero, higher-order perturbative effects induce Rabi oscillations between the ground and excited states. Furthermore, numerical simulations in a time domain including the effect of decoherence demonstrated that coherent oscillations decay, leading to the appearance of PR.
Quantum annealing aims at solving combinatorial optimization problems mapped to Ising interactions between quantum spins. Here, with the objective of developing a noise-resilient annealer, we propose a paradigm for quantum annealing with a scalable network of two-photon-driven Kerr-nonlinear resonators. Each resonator encodes an Ising spin in a robust degenerate subspace formed by two coherent states of opposite phases. A fully connected optimization problem is mapped to local fields driving the resonators, which are connected with only local four-body interactions. We describe an adiabatic annealing protocol in this system and analyse its performance in the presence of photon loss. Numerical simulations indicate substantial resilience to this noise channel, leading to a high success probability for quantum annealing. Finally, we propose a realistic circuit QED implementation of this promising platform for implementing a large-scale quantum Ising machine.
Abstract A Kerr parametric oscillator (KPO) is a Kerr-nonlinear resonator driven parametrically, which can serve as a qubit in quantum information processing. Spectroscopy using a weak probe field provides a practical method for extracting system parameters. In this work, we theoretically investigate reflection and transmission measurements of two-photon and four-photon KPOs, introducing interference effects between inter-level transitions. Because KPOs typically exhibit energy-level degeneracy, a probe field can simultaneously resonate with multiple inter-level transitions. We extend the previous theory of reflection measurements by incorporating the interaction between inter-level transitions and off-diagonal elements of the density matrix which had previously been neglected. We demonstrate that interference among these transitions substantially modifies the spectrum. We identify the conditions for the interference, as well as those under which the off-diagonal elements of the density matrix affect the spectrum. The theory is also generalized to transmission measurements, and is applicable to a broad class of systems beyond KPOs.