Quantum annealing with antiferromagnetic fluctuations

We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic $p$-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of $p$. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of $p$ for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of $p$ efficiently, in contrast to the case with only simple transverse-field fluctuations.

Figure 1

Free energy vs $s$ for some values of $p$. The parameter $\lambda$ is 0.1 (top) and 0.3 (bottom). The dash-dotted line in light green represents the free energy of the QP phase, Eq. (22), the thin solid line in blue is for the F phase, Eq. (23), and the thick solid line in red is for the QP2 phase, Eq. (27). The vertical dashed line denotes the lower limit of the QP2 domain [$s=1/(3-2\lambda )$]. Although it is difficult to discern on the present scale, all the data for finite $p$ that we studied have lower values than that of $f_{\text{QP2}}$ in the QP2 domain.

Figure 2

Magnetization $m^x$ corresponding to Fig. 1. The solid line represents the $x$ component of magnetization of the QP2 phase (24), and the vertical dashed lines are the same as those in Fig. 1. For $\lambda =0.1$ (top panel) and $p⩾5$, a second-order transition occurs at the boundary of the QP2 domain; The magnetization decreases continuously from unity to zero. In contrast, the magnetization for $\lambda =0.3$ (bottom panel) has a jump.

Figure 3

Phase diagrams on the $s$-$\lambda$ plane for $p=3$ (left), $p=5$ (middle), and $p=11$ (right). The dash-dotted line represents the boundary of the QP2 domain [$s=1/(3-2\lambda )$], where a transition takes place between the QP and F phases. For large $\lambda$, the QP and F phases are separated by the horizontal phase boundary (QP-F boundary). The thick solid line in red represents the first-order transition, and the thin solid line in light green is for the second-order transition. For $p=5$ and 11, the magnetization jumps on the dashed line in blue (F-F boundary) within the F phase [21].

Figure 4

Top panel: Energy gap vs $s$ for $p=11$ and $\lambda =0.3$. The vertical dashed lines represent the boundary of the QP2 domain at $s\simeq 0.4167$ and the F-F boundary at $s\simeq 0.4701$. The bottom panel is an enlarged view of the top panel for $N=140$.

Figure 5

The rightmost local minimum of the energy gap as a function of $N$ for $p=11$ and $\lambda =0.3$ on a semilogarithmic scale. The gap closes exponentially with increasing $N$.

Figure 6

Energy gap vs $N$ at local minima for $p=5,\lambda =0.1$ on a log-log scale. We number the minima from left to right. No gaps vanish exponentially up to the size studied here.

Figure 7

Minimum gap vs $N$ for some values of $p$ on a log-log scale for $\lambda =0.1$. Each data set scales polynomially.

Figure 8

Phase diagram in the limit $p\to \infty$. Three lines represent the same phase boundary as those in Fig. 3. The QP phase has the magnetization $m^z=0$. The F phase above the F-F boundary, shown dashed in blue, has the magnetization $m^z=1$, and the phase below the F-F boundary has $0<m^z<1$.