Quantum annealing of an Ising spin-glass by Green’s function Monte Carlo

We present an implementation of quantum annealing (QA) via lattice Green’s function Monte Carlo (GFMC), focusing on its application to the Ising spin glass in transverse field. In particular, we study whether or not such a method is more effective than the path-integral Monte Carlo- (PIMC) based QA, as well as classical simulated annealing (CA), previously tested on the same optimization problem. We identify the issue of importance sampling, i.e., the necessity of possessing reasonably good (variational) trial wave functions, as the key point of the algorithm. We performed GFMC-QA runs using such a Boltzmann-type trial wave function, finding results for the residual energies that are qualitatively similar to those of CA (but at a much larger computational cost), and definitely worse than PIMC-QA. We conclude that, at present, without a serious effort in constructing reliable importance sampling variational wave functions for a quantum glass, GFMC-QA is not a true competitor of PIMC-QA.

Figure 1

(Color online) (Top) Plot of the optimal $\beta$, ${\beta }_{\mathit{opt}}$, for the “Boltzmann" trial wave function ${\psi }_T^{\left(\beta \right)}$ defined in Eq. (4), for several values of $\Gamma$. The dashed line is the fit employed in Sec. 4. (Center) Optimal variational energies ${ϵ}_{\mathit{tot}}^{\left(\mathit{Boltz}\right)}$ corresponding to the ${\beta }_{\mathit{opt}}$ shown in the top panel, and the GFMC estimate of the total energy per spin. The inset magnifies the small-$\Gamma$ region, where little differences are noticeable. (Bottom) The variation residual diagonal energy ${ϵ}_{\mathit{res}}^{\left(\mathit{Boltz}\right)}$ corresponding to the ${\beta }_{\mathit{opt}}$ shown in the top panel, and two GFMC estimators of the residual diagonal energy: the mixed and the Ceperley one (see the text). The dashed horizontal line represents the best residual energy ever achieved, for $\Gamma >0.01$, by employing the mean-field trial wave function in Eq. (3).

Figure 2

(Color online) The average best residual energy obtained by GFMC-QA for the random Ising model instance studied in Refs. 7, 8, versus the total annealing time $\tau$. Upper rhombi: GFMC-QA results without importance sampling $({\psi }_T=1)$. Lower rhombi: GFMC-QA results with importance sampling performed by using the optimal trial wave function ${\psi }_T^{\left(\beta \right)}$ of Sec. 3. The GFMC time unit is a single spin flip, while CA and PIMC-QA Monte Carlo time units are sweeps of the entire lattice (see Ref. 8). The transverse field is linearly reduced down to ${10}^{-4}$ in a total annealing time $\tau$, starting from ${\Gamma }_0=2.5$. Previous results obtained by classical simulated annealing (CA) and by path-integral Monte Carlo quantum annealing (PIMC-QA) [7, 8] are shown for comparison.