Rovibrational controlled-NOT gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory

Implementation of quantum controlled-NOT (CNOT) gates in realistic molecular systems is studied using stimulated Raman adiabatic passage (STIRAP) techniques optimized in the time domain by genetic algorithms or coupled with optimal control theory. In the first case, with an adiabatic solution (a series of STIRAP processes) as starting point, we optimize in the time domain different parameters of the pulses to obtain a high fidelity in two realistic cases under consideration. A two-qubit CNOT gate constructed from different assignments in rovibrational states is considered in diatomic (NaCs) or polyatomic $\left({\text{SCCl}}_2\right)$ molecules. The difficulty of encoding logical states in pure rotational states with STIRAP processes is illustrated. In such circumstances, the gate can be implemented by optimal control theory and the STIRAP sequence can then be used as an interesting trial field. We discuss the relative merits of the two methods for rovibrational computing (structure of the control field, duration of the control, and efficiency of the optimization).

Figure 1

(Color online) Transitions involved in the adiabatic process for computing a CNOT gate by the pulse sequence ${\Omega }_S{\Omega }_{10}{\Omega }_{11}{\Omega }_S$.

Figure 2

(Color online) Upper panel: Potential energy curves of NaCs. The ${{}^1\Sigma }^{+}$ and ${}^1\Pi$ singlet states are in full lines (blue for the ${{}^1\Sigma }^{+}$ states and red for the ${}^1\Pi$ states) and the triplet states ${}^3\Pi$ are in dotted lines. The selected electronic states are the ground $X$ ${{}^1\Sigma }^{+}$ and $B$ ${}^1\Pi$ states (noted by an arrow). Lower panel: Electronic transition dipole moment between the selected $X$ ${{}^1\Sigma }^{+}$ and $B$ ${}^1\Pi$ electronic states.

Figure 3

(Color online) Scheme of the qubit states with $q_c$ in the rotation for the diatomic molecule. $v$ and $J$ are the vibrational and rotational quantum numbers, respectively. $|S⟩$ and $|I⟩$ are the shelving and intermediate states of the STIRAP sequence (see Fig. 1).

Figure 4

(Color online) CNOT gate by APT with the control qubit encoded in the rotational states $J=0$ and $J=2$ and the target qubit in the vibrational states $v=0$ and $v=2$ (see Fig. 3). Upper panel: Evolution of the populations which should remain unaltered when the control qubit is 0, $|01⟩$ $\left(\text{full}\text{line}\right)\to |S^{\prime }⟩$ $\left(\text{dashed line}\right)\to |00⟩$ (dotted line) and $|00⟩$ $\left(\text{full}\text{line}\right)\to |01⟩$ (dotted line). The new shelving state found by the system is $|S^{\prime }⟩=|{{}^1\Sigma }^{+},3,0⟩$ and the new intermediate state is $|I^{\prime }⟩=|{}^1\Pi ,4,1⟩$. Lower panel: Evolution of the populations of the states when the control qubit is 1 (see Fig. 1), $|11⟩$ $\left(\text{full}\text{line}\right)\to |S⟩$ $\left(\text{dashed line}\right)\to |10⟩$ (dotted line) and $|11⟩$ $\left(\text{full}\text{line}\right)\to |10⟩$ (dotted line).

Figure 5

(Color online) Scheme showing the unwanted transitions due to the ${\Omega }_S{\Omega }_{11}{\Omega }_{10}{\Omega }_S$ pulse sequence with assignment 1 (control qubit encoded in the rotational states). The expected transitions from the $|10⟩$ and $|11⟩$ states are drawn in dashed lines. The unwanted transitions from the $|00⟩$ and $|01⟩$ states are drawn in dotted lines (intrusive shelving state $|S^{\prime }⟩=|{{}^1\Sigma }^{+},3,0⟩$ and intrusive intermediate one $|I^{\prime }⟩=|{}^1\Pi ,4,1⟩$).

Figure 6

(Color online) Frequencies in ${\text{cm}}^{-1}$ of the Fourier transform of the OCT field for the CNOT gate with the control qubit in the rotational states $J=0$ and $J=2$ and the target qubit in the vibrational states $v=0$ and $v=2$ (see Fig. 3) for the NaCs system. (Upper panels) First optimization without filtering (a) with the STIRAP sequence as a trial field and (b) with a guess field containing the frequencies ${\Omega }_{11}$ and ${\Omega }_{10}$. (Lower panels) Spectrum of the optimal fields after filtering.

Figure 7

(Color online) CNOT gate with phase constraint [Eq. (2)] optimized by OCT with the control qubit encoded in the rotational states $J=0$ and $J=2$ and the target qubit in the vibrational ones $v=0$ and $v=2$ (see Fig. 3). Population evolution of the four qubit states: $|00⟩$ $\left(|{{}^1\Sigma }^{+},0,0⟩\right)$ in full lines, $|01⟩$ $\left(|{{}^1\Sigma }^{+},2,0⟩\right)$ in dotted line, $|10⟩$ $\left(|{{}^1\Sigma }^{+},0,2⟩\right)$ in dotted line, and $|11⟩$ $\left(|{{}^1\Sigma }^{+},2,2⟩\right)$ in full line. One observes intermediate populations of the state $|S^{\prime }⟩=\left(|{{}^1\Sigma }^{+},3,0⟩\right)$ or $|S⟩$ $\left(|{{}^1\Sigma }^{+},3,2⟩\right)$. The population of the other states of the basis set is not drawn.

Figure 8

(Color online) Scheme of the qubit states with $q_c$ in the vibration for the diatomic example. $v$ and $J$ are the vibrational and rotational quantum numbers, respectively. $|S⟩$ and $|I⟩$ are the shelving and intermediate states of the STIRAP sequence (see Fig. 1)

Figure 9

(Color online) Scheme of the qubit states belonging to four different vibrational states for the diatomic example. $v$ and $J$ are the vibrational and rotational quantum numbers, respectively. $|S⟩$ and $|I⟩$ are the shelving and intermediate states of the STIRAP sequence (see Fig. 1)

Figure 10

(Color online) CNOT gate by the STIRAP sequence with phase constraint [Eq. (2)] when the qubit states are encoded in the rovibrational states $J=0$ of different vibrational states $v=0$, 1, 2, and 3 (see Fig. 9). The initial state is a superposition of all the qubit states with unequal weights. The populations in the states $|00⟩$ $\left(|{{}^1\Sigma }^{+},0,0⟩\right)$ and $|01⟩$ $\left(|{{}^1\Sigma }^{+},1,0⟩\right)$ remain constant. One observes the exchange of the populations in $|11⟩$ $\left(|{{}^1\Sigma }^{+},3,0⟩\right)$ in dotted line and of $|10⟩$ $\left(|{{}^1\Sigma }^{+},2,0⟩\right)$ in dashed line. The shelving state $|S⟩$ is $\left(|{{}^1\Sigma }^{+},4,0⟩\right)$.

Figure 11

(Color online) CNOT gate by the STIRAP sequence with phase constraint [Eq. (2)] and with the qubit states encoded in the rovibrational states $|0_{00}⟩$ of different excited vibrational states of ${\text{SCCl}}_2$ (see Table ). Evolution of the populations in the qubit states $|00⟩$ and $|01⟩$ which remain constant. Evolution of the interchanging populations $|10⟩$ in dashed line and $|11⟩$ in dotted line.