Quantum Annealing and Thermalization: Insights from Integrability

We solve a model that has basic features that are desired for quantum annealing computations: entanglement in the ground state, controllable annealing speed, ground state energy separated by a gap during the whole evolution, and a programmable computational problem that is encoded by parameters of the Ising part of the spin Hamiltonian. Our solution enables exact nonperturbative characterization of final nonadiabatic excitations, including a scaling of their number with the annealing rate and the system size. We prove that quantum correlations can accelerate computations and, at the end of the annealing protocol, lead to the perfect Gibbs distribution of all microstates.

Figure 1

(a) During QA, the entangled ground state is transformed adiabatically into the ground state of the Ising spin Hamiltonian. (b) Evolution of the spectra of the QA Hamiltonian (3) in nonadiabatic ($g=1/N$, top) and nearly adiabatic ($g=1$, bottom) regimes. The ground level is marked by red color. Here, $N=12$, $S_{\text{tot}}^z=0$, ${ϵ}_j=j/N+{\xi }_j$, and ${\xi }_j$ are uniformly distributed random real numbers from the range $\mathbf{(}-1/(2N),1/(2N)\mathbf{)}$. The inset shows exact level crossings indicating the model’s integrability.

Figure 2

(a) The probabilities to remain in the ground state at different $g$ and $N$. Solid curves and the limit $N\to \infty$ (black dashed curve) are predictions of Eq. (12) and point markers are the numerical results [19]. (b) Time dependence of computation accuracy. Solid curves are results of the numerical solution for the Hamiltonian ${\widehat{H}}_{\mathrm{BCS}}(t)$ with $N=12$, $S_{\text{tot}}^z=0$, and the same ${ϵ}_j$ as in Fig. 1.

Figure 3

Two paths corresponding to the same evolution operator. Evolution takes place over the space of real time $t$ and complex values of ${ϵ}_{-}\equiv {ϵ}_j-{ϵ}_{j+1}$. The initial point $a$ corresponds to $t=0_{+}$ and ${ϵ}_j<{ϵ}_{j+1}$. The final point $d$ is at $t\to \infty$ and ${\overline{ϵ}}_j={ϵ}_{j+1}$, ${\overline{ϵ}}_{j+1}={ϵ}_j$. The red path $a\to b\to d$ avoids the singularity at ${ϵ}_{-}=0$ from the infinitesimally small distance $r$ at $t=0_{+}$, and the blue path $a\to c\to d$ avoids this singularity at $t\to \infty$ along the arc $(cd)$ with a finite radius. Evolution over links $(ab)$, $(bd)$, $(ac)$, and $(cd)$ is described by matrices, respectively, $V$, $S^{\overline{\mathit{ϵ}}}$, $S^{\mathit{ϵ}}$, and $V^{\prime }$.

Figure 4

(a) The final polarization of several spins for $N=12$ and $S_{\text{tot}}^z=0$. Prediction of the Gibbs distribution (solid curves) is compared to numerical solution of the Schrödinger equation (point markers) [19]. Here, ${ϵ}_j$ are the same as in Fig. 1. (b) Accuracy of QA at different $g$ and $N$ at $t\to \infty$. Points show exact predictions of Eq. (12), and solid lines are the large-$N$ approximation (13).