Modeling of quantum-information processing with Ehrenfest guided trajectories: A case study with spin-boson-like couplings

We apply a numerical method based on multiconfigurational Ehrenfest trajectories and demonstrate converged results for the Choi fidelity of an entangling quantum gate between two two-level systems interacting through a set of bosonic modes. We consider both spin-boson and rotating-wave Hamiltonians for various numbers of mediating modes (from 1 to 100) and extend our treatment to include finite temperatures. Our results apply to two-level impurities interacting with the same band of a photonic crystal or to two distant ions interacting with the same set of motional degrees of freedom.

Figure 1

Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1$, and $g_2=1.9$, obtained at zero temperature by the dot-dashed curve: MCE method and dotted curve: exact analytic integration for (a) $M=1$ and ${\omega }_1=0.1$ and (b) $M=3$ and ${\omega }_m=0.1m$ for $1\le m\le 3$. The lines $F=0.25$ are reported for reference. All the quantities plotted are dimensionless.

Figure 2

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10$, zero temperature, and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 3

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =10$, and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 4

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =10$, and different values of $g_2/g_1$ (red stands for higher values, and blue stands for lower values). All the quantities plotted are dimensionless.

Figure 5

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =5$, and different values of $g_2/g_1$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 6

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =5$, and different values of $g_2/g_1$ (red stands for higher values, and blue stands for lower values). All the quantities plotted are dimensionless.

Figure 7

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =1,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10$ and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 8

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =1,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10$ and different values of $g_2/g_1$ (red stands for higher values, and blue stands for lower values). All the quantities plotted are dimensionless.

Figure 9

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =1,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =10$, and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 10

MCE results for the Choi fidelity $F$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =1,g_1=1,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =10$, and different values of $g_2/g_1$ (red stands for higher values, and blue stands for lower values). All the quantities plotted are dimensionless.

Figure 11

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for $\widehat{H}$ with $\varepsilon =\Delta =1,g_1=0.5,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10$ and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 12

MCE results for the Choi fidelity $F$ versus rescaled time for $\widehat{H}$ with $\varepsilon =\Delta =1,g_1=0.5,M=10$, and ${\omega }_m=0.1m$ for $1\le m\le 10,\beta =10$, and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 13

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =1,g_1=1,M=20$, and ${\omega }_m=0.1m$ for $1\le m\le 10,{\omega }_m=0.1(m-10)$ for $11\le m\le 20$ and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 14

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for $\widehat{H}$ with $\varepsilon =\Delta =1,g_1=0.5,M=20$, and ${\omega }_m=0.1m$ for $1\le m\le 10,{\omega }_m=0.1(m-10)$ for $11\le m\le 20$ and different values of $g_2$. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 15

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for $\widehat{H}$ with $\varepsilon =\Delta =1$ and a common Ohmic bath with $\alpha =0.09,{\omega }_c=2.5$, and different numbers of total bath modes. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 16

MCE results for the Choi fidelity $F$ versus rescaled time at zero temperature for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =1$ and a common Ohmic bath with $\alpha =0.09$, different cutoff frequencies, and different numbers of total bath modes. The line $F=0.25$ is reported for reference. All the quantities plotted are dimensionless.

Figure 17

MCE results for the concurrence versus rescaled time at different temperatures for ${\widehat{H}}_{\mathrm{rw}}$ with $g_1=1$ and $g_2=2.1$. In (a), $\varepsilon =\Delta =0,M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$), and the initial state is $|4\rangle =|\uparrow \uparrow \rangle$; in (b), $\varepsilon =\Delta =0,M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$), and the initial state is $|2\rangle =|\uparrow \downarrow \rangle$; in (c), $\varepsilon =\Delta =1$, the initial state is $|2\rangle =|\uparrow \uparrow \rangle$, and $M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$) for the dashed-dotted line and $M=20$ [with ${\omega }_m=0.1m$ for $1\le m\le 10$ and ${\omega }_m=0.1(m-10)$ for $11\le m\le 20$] for the dashed line, respectively. All the quantities plotted are dimensionless.

Figure 18

(a) Norm and (b) $E/\Delta$ (where $E$ is the expectation value of energy) for MCE results at zero temperature for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =g_1=1,g_2=2.7,M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$), and different values of $N$ and comp. All the quantities plotted are dimensionless.

Figure 19

Entries of the qubits' density matrix (a) ${\varrho }_{11}$ and (b) ${\varrho }_{13}$ for MCE results at zero temperature for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =g_1=1,g_2=2.7,M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$), and different values of $N$ and comp. The line label is the same as in Figs. 18a and 18b. All the quantities plotted are dimensionless.

Figure 20

(a) Norm and (b) $E/\Delta$ (where $E$ is the expectation value of energy) for MCE results at zero temperature for $\widehat{H}$ with $\varepsilon =\Delta =g_1=g_2=1,M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$), and different values of $N$ and comp. All the quantities plotted are dimensionless.

Figure 21

Entries of the qubits' density matrix (a) ${\varrho }_{11}$ and (b) ${\varrho }_{13}$ for MCE results at zero temperature for $\widehat{H}$ with $\varepsilon =\Delta =g_1=g_2=1,M=10$ (with ${\omega }_m=0.1m$ for $1\le m\le 10$), and different values of $N$ and comp. All the quantities plotted are dimensionless.

Figure 22

(a) Choi fidelity, concurrence for two different separable initial states (b) $|\uparrow \uparrow \rangle$, and (c) $|\uparrow \downarrow \rangle$ versus rescaled time for ${\widehat{H}}_{\mathrm{rw}}$ with $\varepsilon =\Delta =0,g_1=1,g_2=2.1,\beta =0.5$, and different values of $N_T$ and $M$. In all plots, $M=10$ with ${\omega }_m=0.1m$ for $1\le m\le 10$. “Conjugate” refers to the fact that, for those curves, the initial centers of the coherent states ${\mathit{\alpha }}_j$ are in complex-conjugate pairs.