Quantum Computation Based on Quantum Adiabatic Bifurcations of Kerr-Nonlinear Parametric Oscillators

Abstract

Quantum computers with Kerr-nonlinear parametric oscillators (KPOs) have recently been proposed by the author and others. Quantum computation using KPOs is based on quantum adiabatic bifurcations of the KPOs, which lead to quantum superpositions of coherent states, such as Schrödinger cat states. Therefore, these quantum computers are referred to as "quantum bifurcation machines (QbMs)." QbMs can be used for qauntum adiabatic optimization and universal quantum computation. Superconducting circuits with Josephson junctions, Josephson parametric oscillators (JPOs) in particular, are promising for physical implementation of KPOs. Thus, KPOs and QbMs offer not only a new path toward the realization of quantum bits (qubits) and quantum computers, but also a new application of JPOs. Here we theoretically explain the physics of KPOs and QbMs, comparing them with their dissipative counterparts. Their physical implementations with superconducting circuits are also presented.

Analog errors in quantum annealing: doom and hope

“…This reassuring conclusion is reached here using the simplest possible error suppression and correction scheme [25], so that much room for improvement remains for more advanced methods. We expect our results to apply broadly, certainly beyond the D-Wave devices to other quantum [32][33][34] and semiclassical annealing implementations [35,36], and to other forms of analog quantum computing [37].…”

Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems

“…For the Ising problem, a new kind of quantum adiabatic optimization using a network of Kerr-nonlinear parametric oscillators (KPOs) has recently been proposed ( 30 – 34 ). The quantum mechanical Hamiltonian in this approach is given by ( 30 , 34 )Hqfalse(italictfalse)=normalℏtruetrue∑i=1italicNtrue[K2ai†2ai2−italicpfalse(italictfalse)2false(ai†2+ai2false)+Δiai†aitrue]−normalℏξ0truetrue∑i=1italicNtruetrue∑j=1italicNJitalici,italicjai†ajwhere ℏ is the reduced Planck constant, ai† and a i are the creation and annihilation operators, respectively, for the i th oscillator, K is the positive Kerr coefficient, p ( t ) is the time-dependent parametric two-photon pumping amplitude, Δ i is the positive detuning frequency between the resonance frequency of the i th oscillator and half the pumping frequency, and ξ 0 is a positive constant with the dimension of frequency. The initial state of each KPO is set to the vacuum state, and the pumping amplitude p ( t ) is gradually increased from zero to a sufficiently large value.…”

“…The initial state of each KPO is set to the vacuum state, and the pumping amplitude p ( t ) is gradually increased from zero to a sufficiently large value. The constant ξ 0 is set to a sufficiently small value such that the vacuum state is the ground state of the initial Hamiltonian ( 30 , 34 ). Then, each KPO finally becomes a coherent state with a positive or negative amplitude via a quantum adiabatic bifurcation ( 30 , 34 ), and the sign of the final amplitude for the i th KPO provides the i th Ising spin of the ground state of the Ising model, which is guaranteed by the quantum adiabatic theorem ( 30 , 34 ).…”

“…The corresponding classical Hamiltonian system is derived from a classical approximation where the expectation value of a i is approximated as a complex amplitude x i + iy i (where i denotes the imaginary unit) ( 30 , 34 ). Here, the real and imaginary parts, x i and y i , are a pair of canonical conjugate variables, such as a position and a momentum, for the i th KPO.…”

“…Inspired by recently proposed quantum adiabatic optimization using a nonlinear oscillator network ( 30 – 34 ), here, we propose a new heuristic algorithm for the Ising problem. In this algorithm, which we call simulated bifurcation (SB), we numerically simulate adiabatic evolutions of classical nonlinear Hamiltonian systems exhibiting bifurcation phenomena ( 35 ), where two branches of the bifurcation in each nonlinear oscillator correspond to two states of each Ising spin ( 30 , 34 ). [The CIMs also use two branches of a bifurcation for two states of an Ising spin ( 15 – 20 ), but they are not Hamiltonian systems.]…”