Quantum annealing with all-to-all connected nonlinear oscillators

Abstract

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.

Figure 1. Contour plot of the metapotential.

Metapotential corresponding to , where K is the Kerr-nonlinearity, and are the strengths of the two-photon and single-photon drive respectively, with and (a) , (b) . The metapotentials, shown in the units of the Kerr-nonlinearity K, are characterized by (a) two peaks of equal heights corresponding to the degenerate states and , and (b) two peaks of different heights, indicating lifting of degeneracy between the encoded spin states and .

Figure 2. Adiabatic protocol with single spin.

(a) Change of the energy of the ground and first excited state as a function of time in a single resonator for , and δ₀=0.2K. The minimum energy gap is also shown with Δ_min=0.16K. (b) The Wigner function of the KNR state at three different times when initialized to either the excited or (vacuum) ground state , respectively. (c) Metapotential corresponding to with and showing two peaks of unequal height. The lower peak (corresponding to the ground state) is circular, whereas the higher one (corresponding to the excited state) is deformed as highlighted by black circles. (d) Transition matrix elements between the ground and excited states in the event of a photon jump during the adiabatic protocol.

Figure 3. Success probability for the two coupled spins problem.

Loss-rate dependence of the success probability for the two-spin adiabatic algorithm in a system of two-photon-driven KNRs with single-photon loss κ (green squares) and qubits with pure dephasing at rate γ_φ (red squares). The quality factor Q=ω_r/κ is indicated on the top axis for a KNR of frequency ω_r/2π=5 GHz.

Figure 4. Physical realization of the LHZ scheme.

(a) Illustration of the plaquette consisting of four JPAs coupled by a Josephson junction (JJ). The four JPAs have different frequencies (indicated by colours) and are driven by two-photon drives such that ω_p,k+ω_p,l=ω_p,m+ω_p,n. The nonlinearity of the JJ induces a four-body coupling between the KNRs. (b) Illustration of a fully connected Ising problem with N=5 logical spins. (c) The same problem embedded on M=10 physical spins and 3 fixed spins on the boundary.

Figure 5. Success probability for the frustrated three-spin problem.

(a) With LHZ encoding: Probability of successfully finding the ground state of a frustrated three-spin Ising problem by implementing the adiabatic algorithm on a plaquette of four KNRs with single-photon loss (green squares) for , δ₀=0.45K, C=0.05K, J=0.095K. The success probability for an implementation with qubits with pure dephasing rate γ_φ is also shown (red squares). The two cases are designed to have identical Δ_min and computation time τ=40/Δ_min. The quality factor Q=ω_r/κ is indicated on the top axis for a KNR of frequency ω_r/2π=5 GHz. (b) Without encoding: Probability of successfully finding the ground state of a frustrated three-spin Ising problem by implementing the adiabatic algorithm on three directly coupled KNRs with single-photon loss (green squares) for , δ₀=0.45K, J_k,j=0.095K for k,i=1, 2, 3. Note that the local drive J in the embedded problem is same as the coupling J_k,j in the un-embedded one and the minimum energy gap in the un-embedded problem is twice that of the embedded problem. The success probability for an implementation with qubits without encoding and with pure dephasing is also shown (red squares).