Direct comparison of quantum and simulated annealing on a fully connected Ising ferromagnet

Abstract

We compare the performance of quantum annealing (QA, through Schrödinger dynamics) and simulated annealing (SA, through a classical master equation) on the p-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second-order (p = 2, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition (p ≥ 3, i.e., with multi-spin Ising interactions) , in contrast, the classical annealing dynamics appears to remain stuck in the disordered phase, while we have clear evidence that QA shows a residual energy which decreases towards 0 when the total annealing time τ increases, albeit in a rather slow (logarithmic) fashion. This is one of the rare examples where a limited quantum speedup, a speedup by QA over SA, has been shown to exist by direct solutions of the Schrödinger and master equations in combination with a non-equilibrium Landau-Zener analysis. We also analyse the imaginary-time QA dynamics of the model, finding a 1/τ 2 behaviour for all finite values of p, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit p(odd) = ∞ is also discussed.

Quantum annealing speedup over simulated annealing on random Ising chains

“…Hence, provocatively, "quantum inspired" is here better than "quantum" , a point that deserves further studies. Results on the fully-connected Ising ferromagnet confirm that this IT-speedup is not specific to the present 1d problem 59 .…”

Lindblad dissipative dynamics in the presence of phase coexistence

“…In all three cases the model undergoes a phase transition increasing either temperature T or transverse field h x from a phase with finite to one with vanishing expectation value of σ z i , ∀ i. This transition is first order in case 1 38,45 and 2 46 , but second order in case 3 36 . Moreover, in case 1 and 3 it corresponds to the restoration of the Z 2 symmetry σ z i → −σ z i , ∀ i, which is spontaneously broken in the low T and h x phase, while such symmetry is explicitly broken in case 2.…”

Relaxation of the Ising spin system coupled to a bosonic bath and the time dependent mean field equation

“… where This master equation has only N + 1 variables instead of 2 N , thus is easy to simulate for large systems. A comparison between the quantum and the thermal simulated annealing of the fully connected Ising model was investigated by Wauters et al using a similarly reduced master equation [ 42 ]. Eq (27) has the form and we want to determine the lowest (nonzero) eigenvalue of M red , which is the same as the lowest (nonzero) eigenvalue of M .…”