Simple Glass Models and Their Quantum Annealing

We study first-order quantum phase transitions in mean-field spin glasses. We solve the quantum random energy model using elementary methods and show that at the transition the eigenstate suddenly projects onto the unperturbed ground state and that the gap between the lowest states is exponentially small in the system size. We argue that this is a generic feature of all “random first-order” models, which includes benchmarks such as random satisfiability. We introduce a two-time instanton to calculate this gap in general, and discuss the consequences for quantum annealing.

Figure 1

Left: Evolution of lowest energy levels for a single realization of the QREM with $N=20$ spins (dots) compared with analytical predictions (lines). Inset: Evolution of the ensemble averaged minimal gap at the transition. Center: Phase diagram of the QREM in temperature $T$ and transverse field $\Gamma$. At $T=0$ the quantum transition arises at ${\Gamma }_c=\sqrt{\ln 2}$ while the classical glass transition for $\Gamma =0$ is at $T_c=\sqrt{\ln 2}/2$. Right: A multi-instanton configuration for the two-times overlap $q_{t,t^{\prime }}$ and the two-time Lagrange multipliers ${\tilde{q}}_{t,t^{\prime }}$. Far from the jump times, the functions $q_{t,t^{\prime }}$ and ${\tilde{q}}_{t,t^{\prime }}$ take the same form as those computed at those times for the glass phase [the regions $(1,1)$], for the quantum paramagnet [in the regions $(2,2)$], and are zero in the mixed regions $(1,2)\mathrm{\text{−}}(2,1)$. In the large $p$ limit the problem can be solved completely using the so-called “static approximation” [5, 14] within the $(1,1)$ regions, and, in addition, the fact that ${\tilde{q}}_{t,t^{\prime }}^d$ and ${\tilde{q}}_{t,t^{\prime }}$ become either infinity or zero, with sharp interfaces.