Quantum annealing of the $p$-spin model under inhomogeneous transverse field driving

We solve the mean-field-like $p$-spin Ising model under a spatiotemporal inhomogeneous transverse field to study the effects of inhomogeneity on the performance of quantum annealing. We previously found that the problematic first-order quantum phase transition that arises under the conventional homogeneous field protocol can be avoided if the temperature is zero and the local field is completely turned off site by site after a finite time. We show in the present paper that, when these ideal conditions are not satisfied, another series of first-order transitions appear, which prevents us from driving the system while avoiding first-order transitions. Nevertheless, under these nonideal conditions, quantitative improvements can be obtained in terms of narrower tunneling barriers in the free-energy landscape. A comparison with classical simulated annealing establishes a limited quantum advantage in the ideal case, since inhomogeneous temperature driving in simulated annealing cannot remove a first-order transition, in contrast to the quantum case. The classical model of spin-vector Monte Carlo is also analyzed, and we find it to have the same thermodynamic phase diagram as the quantum model in the ideal case, with deviations arising at nonzero temperature.

Figure 1

Phase diagram on the $s-\tau$ plane for the idealized case at zero temperature. Each color denotes a line of first-order transitions for a given $p$, which is chosen to be 3, 4, and 5.

Figure 2

Two types of energy gap ${\mathrm{\Delta }}_{a_1}$ and ${\mathrm{\Delta }}_b$ for $p=3$ as functions $s$ for (a) $\tau =s$ (away from the transition line) and (b) $\tau =s^{2.366}$ (just touching the critical point). The smaller of these two is the final energy gap. (c) The energy gap for finite-size systems with $\tau =s$ obtained by direct numerical diagonalization. The location of the minimum is indicated by an arrow for each $N$.

Figure 3

(a) The dashed lines show the schedule of $\tau$ expressed by Eq. (12) in the phase diagram. The black sold line represents first-order phase transitions. (b) The minimal value of the energy gap against $N$ in a log-log scale as calculated by numerical diagonalization. (c) The energy gap for $N=70$ in two cases $a=0.7$ and 0.8 of Eq. (12). The inset shows the behavior around the phase-transition point. All results shown are for $p=3$.

Figure 4

Two nonideal cases: (a)–(c) finite temperature and (d)–(f) incomplete turn-off . (a) Illustrative behavior of the free energy $f\left(m\right)$ and the jump $\mathrm{\Delta }m$ in the order parameter at a first-order phase transition. (b) Finite-temperature phase diagram for $p=3$. All curves represent first-order phase transitions. The red circle and blue square correspond to the respective points in panel (c). (c) Jump in magnetization along the line of first-order transitions depicted in panel (b). Symbols in red circle and blue square represent the respective points in the phase diagram of panel (b). (d) Amplitude of the transverse field ${\mathrm{\Gamma }}_i$ of Eq. (13). (e) Phase diagram for $p=3$. The curves represent first-order phase transitions. (f) Jump in magnetization $\mathrm{\Delta }m$ along the first-order transition line. We note that (e) and (f) are remarkably similar to (b) and (c), though we do not presently have an explanation for this fact.

Figure 5

(a) The field amplitude ${\mathrm{\Gamma }}_i(\tau ;a)$ of Eq. (15). (b) Phase diagram for $p=3$ for several values of $a$.

Figure 6

(a) Phase diagram for bimodal random longitudinal fields with strengths $h_0=0.1,0.5$, and 1. (b) Phase diagram for Gaussian random longitudinal fields with standard deviations $\sigma =0.1,0.5$, and 1. All lines are for first-order phase transitions. All the data are for $p=3$.

Figure 7

Behavior of the order parameter in simulated annealing with inhomogeneous temperature driving. We choose $p=3$ and ${\beta }_0=2$ and the bimodal distribution of random local fields.

Figure 8

Phase diagram for $p=3$ for the SVMC model.

Figure 9

Left top panel is the phase diagram, where the solid line represents a line of first-order phase transitions, and three lines (a), (b), and (c) indicate paths with $\tau =s,s^{2.366}$, and 0. Panels (a)–(c) show the entanglement entropy for the corresponding paths. In each case we set $p=3$ and $u=1/2$.