Fast forward of the adiabatic spin dynamics of entangled states

We develop a fast-forward scheme of the adiabatic spin dynamics of quantum entangled states. We settle the quasiadiabatic dynamics by adding the regularization terms to the original Hamiltonian and then accelerate it with the use of a large time-scaling factor. Assuming the experimentally realizable candidate Hamiltonian consisting of the exchange interactions and magnetic field, we solve the regularization terms. These terms, multiplied by the velocity function, give rise to the state-dependent counterdiabatic terms. The scheme needs neither knowledge of full spectral properties of the system nor solving the initial- and boundary-value problem. Our fast forward Hamiltonian generates a variety of state-dependent counterdiabatic terms for each of adiabatic states, which can include the state-independent one. We highlight this fact by using minimum (two-spin) models for a simple transverse Ising model, quantum annealing, and generation of entanglement.

Figure 1

Time dependence of three solutions for the state-dependent counterdiabatic interactions: (a) $B_y\left(t\right)=v\left(t\right){\tilde{B}}_y\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (solid line) and $W_2\left(t\right)=v\left(t\right){\tilde{W}}_2\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dashed line), (b) $B_y\left(t\right)=v\left(t\right){\tilde{B}}_y\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (solid line) and $W_1\left(t\right)=v\left(t\right){\tilde{W}}_1\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dashed line), and (c) $W_1\left(t\right)=v\left(t\right){\tilde{W}}_1\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (solid line) and $W_2\left(t\right)=v\left(t\right){\tilde{W}}_2\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dashed line).

Figure 2

Time dependence of $|C_1^{\text{FF}}{|}^2$ (solid line), $|C_2^{\text{FF}}{|}^2$ (dotted line), $|C_3^{\text{FF}}{|}^2$ (dotted line), and $|C_4^{\text{FF}}{|}^2$ (dashed line) obtained (a) by solving the TDSE and (b) from the eigenvector.

Figure 3

Time dependence of two solutions for the state-dependent counterdiabatic interactions: (a) $B_y\left(t\right)=v\left(t\right){\tilde{B}}_y\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (solid line), $W_1\left(t\right)=v\left(t\right){\tilde{W}}_1\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dotted line), and $W_3\left(t\right)=v\left(t\right){\tilde{W}}_3\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dashed line) and (b) $B_x\left(t\right)=v\left(t\right){\tilde{B}}_x\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (solid line), $W_1\left(t\right)=v\left(t\right){\tilde{W}}_1\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dotted line), and $W_2\left(t\right)=v\left(t\right){\tilde{W}}_2\left(R(\mathrm{\Lambda }\left(t\right))\right)$ (dashed line).

Figure 4

Time dependence of $|C_4^{\text{FF}}{|}^2$ (solid line), $|C_2^{\text{FF}}{|}^2$ (dashed line), $|C_3^{\text{FF}}{|}^2$ (dashed line), and $|C_1^{\text{FF}}{|}^2$ (dotted line) obtained (a) by solving the TDSE and (b) from eigenvectors.