Fast and robust control of two interacting spins

Rapid preparation, manipulation, and correction of spin states with high fidelity are requisite for quantum information processing and quantum computing. In this paper, we propose a fast and robust approach for controlling two spins with Heisenberg and Ising interactions. By using the concept of shortcuts to adiabaticity, we first inverse design the driving magnetic fields for achieving fast spin flip or generating the entangled Bell state, and further optimize them with respect to the error and fluctuation. In particular, the designed shortcut protocols can efficiently suppress the unwanted transition or control error induced by anisotropic antisymmetric Dzyaloshinskii-Moriya exchange. Several examples and comparisons are illustrated, showing the advantages of our methods. Finally, we emphasize that the results can be naturally extended to multiple interacting spins and other quantum systems in an analogous fashion.

Figure 1

Schematic diagram for the two-spin system with the Heisenberg or Ising interaction $J$ and Dzyaloshinskii-Moriya interaction $D$, resulting from anisotropic antisymmetric exchange.

Figure 2

Optimal shortcut for spin flip in two Heisenberg-interacting spins with rotating magnetic fields, by taking into account the systematic error. (a) Optimal protocol of two components of magnetic field, $\mathrm{\Omega }$ (red, solid line) and $\mathrm{\Delta }$ (blue, dashed line), in the unit of $B_0$. (b) Dynamics of spin, described by the probability of state $|{\psi }_{1,1}\rangle$ (blue, dashed line), $|{\psi }_{1,0}\rangle$ (black, dotted line), and $|{\psi }_{1,-1}\rangle$ (red, solid line). (c) Fidelity vs the systematic error $\delta$ in the amplitude of rotating magnetic field, where the optimal shortcut (red, solid line) and flat $\pi$ pulse (blue, dashed line) are compared. Parameters: $T=1$ and $\alpha =0.125$.

Figure 3

Optimal shortcut for spin flip in two Heisenberg-interacting spins with rotating magnetic fields, by taking into account the perturbative DM interaction. (a) Optimal protocol of two components of magnetic field, $\mathrm{\Omega }$ (red, solid line) and $\mathrm{\Delta }$ (blue, dashed line), in the unit of $B_0$. (b) Dynamics of spin, described by the probability of state $|{\psi }_{1,1}\rangle$ (blue, dashed line), $|{\psi }_{1,0}\rangle$(black, dotted line), $|{\psi }_{0,0}\rangle$ (purple, dot-dashed line), and $|{\psi }_{1,-1}\rangle$ (red, solid line). (c) Fidelity vs constant vector $D$, representing the magnitude of such anisotropic antisymmetric exchange, where the optimal shortcut (red, solid line) and flat $\pi$ pulse (blue, dashed line) are compared. Parameters: $J/B_0=10, D/B_0=1, T=1$, and $\alpha =0.059$.

Figure 4

Optimal designed protocol for fast generation of the entangled Bell state in two Ising-interacting spins with driving magnetic fields, by taking into account the amplitude error. (a) Optimal protocol of time dependence $\mathrm{\Omega }$ (red, solid line) and $\mathrm{\Delta }$ (blue, dashed line), in the unit of $B_0$. (b) Dynamics of spin, described by the probability of state $|{\psi }_{1,1}\rangle$ (blue, dashed line) and $|{\psi }_{1,0}\rangle$ (red, solid line). (c) Fidelity vs the systematic error $\delta$, comparing the optimal shortcut (red, solid line) with the flat $\pi$ pulse (blue, dashed line). Parameter: $T=1$.

Figure 5

Optimal designed protocol for fast generation of the entangled Bell state in two Ising-interacting spins with driving magnetic fields, by taking into account the DM interaction. (a) Optimal protocol of time dependence $\mathrm{\Omega }$ (red, solid line) and $\mathrm{\Delta }$ (blue, dashed line), in the unit of $B_0$. (b) Dynamics of spin, described by the probability of state $|{\psi }_{1,1}\rangle$ (blue, dashed line) and $|{\psi }_{1,0}\rangle$ (red, solid line). (c) Fidelity vs the systematic error $\delta$, relevant to the DM term, comparing the optimal shortcut (red, solid line) with the flat $\pi$ pulse (blue, dashed line). Parameters: $J/B_0=10, D/B_0=1, T=1$, and $\alpha =-0.206$.

Figure 6

Robustness of designed shortcut protocols (red, solid line), by comparing with flat $\pi$ pulse (black, dotted line) and composite pulse (blue, dashed line), where (a) Heisenberg interaction and (b) Ising interaction are considered. Parameters: $T=1$ and others are the same as those in Figs. 2 and 4.