Experimental implementation of quantum gates through actuator qubits

Universal quantum computation requires the implementation of arbitrary control operations on the quantum register. In most cases, this is achieved by external control fields acting selectively on each qubit to drive single-qubit operations. In combination with a drift Hamiltonian containing interactions between the qubits, this allows the implementation of arbitrary gate operations. Here, we demonstrate an alternative scheme that does not require local control for all qubits: we implement one- and two-qubit gate operations on a set of target qubits indirectly, through a combination of gates on directly controlled actuator qubits with a drift Hamiltonian that couples actuator and target qubits. Experiments are performed on nuclear spins, using radio-frequency pulses as gate operations and magnetic-dipole couplings for the drift Hamiltonian.

Figure 1

Structures and Hamiltonian constants of the molecules used as quantum registers. The actuator and target qubits are marked by solid and dashed rectangles, respectively. (a) 4-fluoro-7-nitro-2,1,3-benzoxadiazole. The fluorine spin F is used as the actuator qubit 1, and the proton spins H1 and H2 are the target qubits 2 and 3. (b) Hamiltonian parameters of molecule (a) in frequency units (Hz): the diagonal elements are the chemical shifts in a 11.7 T field, the off-diagonal terms represent the dipolar coupling constants. The liquid crystal solvent is 1-(trans-4-hexylcyclohexyl)-4-isothiocyanatobenzene. (c) 1,2-difluoro-4-iodobenzene. The proton spins H1–H3 are the actuator qubits 1–3, and the fluorine spins F1 and F2 are the target qubits 4 and 5. (d) The Hamiltonian parameters of the molecule (c). The liquid crystal solvent is ZLI- 1132.

Figure 2

(a) ${}^{19}\mathrm{F}$ and (b) ${}^1\mathrm{H}$ -NMR spectra of 4-fluoro-7-nitro-2,1,3-benzoxadiazole obtained by hard $\pi /2$ pulses from the thermal state, respectively, where the black curves indicate the spectra obtained in experiment and the red by simulation. We shifted the spectra by simulation for easier comparison with the spectra in experiment.

Figure 3

(a) ${}^{19}\mathrm{F}$ and (b) ${}^1\mathrm{H}$ -NMR spectra of 1,2-difluoro-4-iodobenzen obtained by hard $\pi /2$ pulses from the thermal state respectively, where the black curves indicate the spectra obtained in experiment and the red by simulation. (b) Peaks indicated with an asterisk come from the acetone in the coaxis capillary that is used for locking the field.

Figure 4

Fidelities reached in simulated optimized gate operations as a function of the pulse duration for $U_{z,2}\left(\theta \right)=e^{i\theta Z_2/2}$, with $\theta =-\pi /6$ and $-\pi /2$. Fidelities above 0.99 are reached for specific pulse durations. The panels in the bottom part of the figure show the corresponding fidelities for the case of full control of all qubits.

Figure 5

Pulse duration changing with the rotation angle $\theta$ for implementing $U_{z,2}\left(\theta \right)=e^{i\theta Z_2/2}$ with the calculated fidelity higher than 0.99.

Figure 6

Experimental ${}^1\mathrm{H}$-NMR spectra of 4-fluoro-7-nitro-2,1,3-benzoxadiazole demonstrating the implementation of quantum gates. The blue curves in (a)–(e) show reference spectra for the states $\mathit{\text{EYE}}+\mathit{\text{EEY}},\mathbf{10}X,\mathbf{1}X\mathbf{0},\mathbf{10}Y$, and $\mathbf{1}Y\mathbf{0}$, respectively. The red curves show the results of the implementation of the gates $U_z(-\pi )$ to $\mathit{\text{EYE}}+\mathit{\text{EEY}},U_{z,3}(-\pi /2)$ to $-\mathbf{1}\mathbf{0}Y,U_{23}(\pi /4)$ to $-\mathbf{1}\mathbf{0}Y,U_{23}(\pi /4)$ to $\mathbf{1}X\mathbf{0}$, and $U_{z,2}(-\pi /2)$ to $\mathbf{1}X\mathbf{0}$. In (b)–(e) we shifted the red spectra for easier comparison with the reference spectra.

Figure 7

Experimental results of implementing $U_{23}\left(\theta \right)$ in the three-qubit system for the input states $\mathbf{1}X\mathbf{0},-\mathbf{1}Y\mathbf{0},\mathbf{10}X,-\mathbf{10}Y$, shown as panels (a)–(d), respectively. The basis operators for which the overlaps are determined are given in each panel. The solid and dashed curves are fits to the experimental data points.

Figure 8

Experimental ${}^{19}\mathrm{F}$-NMR spectra of 1,2-difluoro-4-iodobenzene, demonstrating the performance of operation $U_{45}(\pi /4)$ in the five-qubit system. The spectra represented as the blue curves in panels (a)–(c) are obtained from the states $\mathit{\text{EEEXE}}+\mathit{\text{EEEEX}},\mathbf{000}X\mathbf{0}$, and $\mathbf{0000}X$, respectively. The spectra represented as the red curves in panels (b)–(c) result from the implementation of $U_{45}(\pi /4)$ to $-\mathbf{0000}Y$ and $-\mathbf{000}Y\mathbf{0}$.

Figure 9

$\chi$ matrices obtained by quantum process tomography for $e^{-i\pi Z/4}$ for the ideal gate and for the experimental implementations on qubits 2 and 3.