Shortcuts to adiabaticity for quantum annealing

We study the Ising Hamiltonian with a transverse field term to simulate the quantum annealing. Using shortcuts to adiabaticity, we design the time dependence of the Hamiltonian. The dynamical invariant is obtained by the mean-field ansatz, and the Hamiltonian is designed by the inverse engineering. We show that the time dependence of physical quantities such as the magnetization is independent of the speed of the Hamiltonian variation in the infinite-range model. We also show that rotating transverse magnetic fields are useful to achieve the ideal time evolution.

Figure 1

Time dependence of the coefficients of the transverse Ising model (Schedule 1).

Figure 2

Time dependence of the coefficients of the transverse Ising model (Schedule 2).

Figure 3

Magnetization for linear schedule. We take $N=4000$, ${\mathrm{\Gamma }}_0=1.0$, and $h=0.1$.

Figure 4

Magnetization for Schedules 1 (top) and 2 (bottom). We take $N=4000$, $J=1.0$, ${\mathrm{\Gamma }}_0=1.0$, and $h=0.1$.

Figure 5

Bloch vector for Schedule 1.

Figure 6

Time evolution of the magnetization for the Hamiltonian with perturbations. We set $(h_0,h_p)=(4.0,0.0)$ (stable) and $(h_0,h_p)=(0.0,4.0)$ (unstable).

Figure 7

Time evolution of the variance of the magnetization for the Hamiltonian with perturbations. We set $(h_0,h_p)=(4.0,0.0)$ (stable) and $(h_0,h_p)=(0.0,4.0)$ (unstable).

Figure 8

Time dependence of the coefficients of the transverse Ising model. We take ${\mathrm{\Gamma }}_0=1.0$ and $J=1.0$.

Figure 9

Time dependence of angles for Schedules 1 (top) and 2 (bottom) in single-spin system. We take ${\mathrm{\Gamma }}_0=1.0$ and $h_1=1.0$.

Figure 10

Magnetic field for Schedule 1 in the single-spin system.

Figure 11

Magnetic field for Schedule 2 in the single-spin system.

Figure 12

Time dependence of angles for Schedules 1 (top) and 2 (bottom) in transverse field Ising model (21). We take ${\mathrm{\Gamma }}_0=1.0$ and $h=0.1$.

Figure 13

Time dependence of angles. We take ${\mathrm{\Gamma }}_0=1.0$ and $J=1.0$.