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arXiv:2005.13819 · npj Quantum Information

High-accuracy Ising machine using Kerr-nonlinear parametric oscillators with local four-body interactions

Taro Kanao, Hayato Goto

Published Jan 29, 2021• 43 citations
arXivHTMLPDF

Abstract

Abstract A two-dimensional array of Kerr-nonlinear parametric oscillators (KPOs) with local four-body interactions is a promising candidate for realizing an Ising machine with all-to-all spin couplings, based on adiabatic quantum computation in the Lechner–Hauke–Zoller (LHZ) scheme. However, its performance has been evaluated only for a symmetric network of three KPOs, and thus it has been unclear whether such an Ising machine works in general cases with asymmetric networks. By numerically simulating an asymmetric network of more KPOs in the LHZ scheme, we find that the asymmetry in the four-body interactions causes inhomogeneity in photon numbers and hence degrades the performance. We then propose a method for reducing the inhomogeneity, where the discrepancies of the photon numbers are corrected by tuning the detunings of KPOs depending on their positions, without monitoring their states during adiabatic time evolution. Our simulation results show that the performance can be dramatically improved by this method. The proposed method, which is based on the understanding of the asymmetry, is expected to be useful for general networks of KPOs in the LHZ scheme and thus for their large-scale implementation.

DOI
10.1038/s41534-020-00355-1

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Abstract A two-dimensional array of Kerr-nonlinear parametric oscillators (KPOs) with local four-body interactions is a promising candidate for realizing an Ising machine with all-to-all spin couplings, based on adiabatic quantum computation in the Lechner–Hauke–Zoller (LHZ) scheme. However, its performance has been evaluated only for a symmetric network of three KPOs, and thus it has been unclear whether such an Ising machine works in general cases with asymmetric networks. By numerically simulating an asymmetric network of more KPOs in the LHZ scheme, we find that the asymmetry in the four-body interactions causes inhomogeneity in photon numbers and hence degrades the performance. We then propose a method for reducing the inhomogeneity, where the discrepancies of the photon numbers are corrected by tuning the detunings of KPOs depending on their positions, without monitoring their states during adiabatic time evolution. Our simulation results show that the performance can be dramatically improved by this method. The proposed method, which is based on the understanding of the asymmetry, is expected to be useful for general networks of KPOs in the LHZ scheme and thus for their large-scale implementation.
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.
Kerr parametric oscillators (KPOs) can stabilize the superpositions of coherent states, which can be utilized as qubits, and are promising candidates for realizing hardware-efficient quantum computers. Although elementary gates for universal quantum computation with KPO qubits have been proposed, these gates are usually based on adiabatic operations and thus need long gate times, which result in errors caused by photon loss in KPOs realized by, e.g., superconducting circuits. In this work we accelerate the elementary gates by experimentally feasible control methods, which are based on numerical optimization of pulse shapes for shortcuts to adiabaticity. By numerical simulations, we show that the proposed methods can achieve speedups compared to adiabatic ones by up to six times with high gate fidelities of 99.9%. These methods are thus expected to be useful for quantum computers with KPOs.
There is a significant ongoing effort in realizing quantum annealing with different physical platforms. The challenge is to achieve a fully programmable quantum device featuring coherent adiabatic quantum dynamics. Here we show that combining the well-developed quantum simulation toolbox for Rydberg atoms with the recently proposed Lechner-Hauke-Zoller (LHZ) architecture allows one to build a prototype for a coherent adiabatic quantum computer with all-to-all Ising interactions and, therefore, a platform for quantum annealing. In LHZ an infinite-range spin-glass is mapped onto the low energy subspace of a spin-1/2 lattice gauge model with quasi-local four-body parity constraints. This spin model can be emulated in a natural way with Rubidium and Caesium atoms in a bipartite optical lattice involving laser-dressed Rydberg-Rydberg interactions, which are several orders of magnitude larger than the relevant decoherence rates. This makes the exploration of coherent quantum enhanced optimization protocols accessible with state-of-the-art atomic physics experiments.
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 Letter is directly relevant for quantum systems operating on coherent states.
Kerr parametric oscillators (KPOs), which can be implemented with superconducting parametrons possessing large Kerr nonlinearity, have been attracting much attention in terms of their applications to quantum annealing, universal quantum computation, and studies of quantum many-body systems. It is of practical importance for these studies to realize fast and accurate tunable coupling between KPOs in a simple manner. We develop a simple scheme of fast tunable coupling of KPOs with high tunability in speed and amplitude using the fast transitionless rotation of a KPO in the phase space based on the shortcuts to adiabaticity. Our scheme enables rapid switching of the effective coupling between KPOs, and can be implemented with always-on linear coupling between KPOs, by controlling the phase of the pump field and the resonance frequency of the KPO without controlling the amplitude of the pump field nor using additional drive fields and couplers. We apply the coupling scheme to a two-qubit gate, and show that our scheme realizes high gate fidelity compared with a purely adiabatic one, by mitigating undesired nonadiabatic transitions.
Ising machines are physical platforms designed to minimize the energy of classical Ising Hamiltonians, yet accessing specific excited states remains an open challenge of both fundamental and practical relevance. In this letter we show that a network of Kerr parametric oscillators (KPOs) naturally implements an Ising selector machine. By tuning the frequency detuning between the parametric pump and the oscillator resonances, the system can be steered to converge close to the ground state, the highest-energy configuration, or targeted intermediate excited states. Beyond mean field, numerical simulations based on the truncated Wigner approximation demonstrate that noise insertion preserves the energetic structure of the landscape. The targeted state emerges with an exponentially enhanced probability over the rest of the Ising spectrum. Our results establish the pump-cavity detuning as a control knob for navigating the full Ising energy landscape, opening a route to applications in Boltzmann sampling, hardness characterization, and spectral analysis of combinatorial problems.