Efficient Quantum Gates for Individual Nuclear Spin Qubits by Indirect Control

Hybrid quantum registers, such as electron-nuclear spin systems, have emerged as promising hardware for implementing quantum information and computing protocols in scalable systems. Nevertheless, the coherent control of such systems still faces challenges. Particularly, the lower gyromagnetic ratios of the nuclear spins cause them to respond slowly to control fields, resulting in gate times that are generally longer than the coherence time of the electron. Here, we demonstrate a scheme for circumventing this problem by indirect control: we apply a small number of short pulses only to the electron and let the full system undergo free evolution under the hyperfine coupling between the pulses. Using this scheme, we realize robust quantum gates in an electron-nuclear spin system, including a Hadamard gate on the nuclear spin and a controlled-NOT gate with the nuclear spin as the target qubit. The durations of these gates are shorter than the electron coherence time, and thus additional operations to extend the system coherence time are not needed. Our demonstration serves as a proof of concept for achieving efficient coherent control of electron-nuclear spin systems, such as nitrogen vacancy centers in diamond. Our scheme is still applicable when the nuclear spins are only weakly coupled to the electron.

Figure 1

MW pulse sequence to realize $U_T$ by IC, at a fixed ${\omega }_1$. The delays ${\tau }_i$, MW pulse durations $t_i$ and phases ${\varphi }_i$ are the free variables to be optimized.

Figure 2

(a) Quantum circuit to test $U_H$. The MW pulse sequence parameters for $U_H$ are given in Table 1. The clean-up operation is represented by the dotted box. (b) Populations (solid circles) and coherences (zigzag arrows) at each stage of the pulse sequence in (a). (c),(d) ${}^{13}\mathrm{C}$ spin spectra obtained by the pulse sequence in (a). (c) Without the first $U_H$. (d) With both $U_H$. Inset: Final population of $|0\uparrow ⟩$ as a function of $t$.

Figure 3

Quantum circuits to test $U_{\mathrm{CNOT}}$. The MW pulse sequence parameters for $U_{\mathrm{CNOT}}$ indicated by red empty boxes are given in Table. 1. ${\theta }_{x/y/\varphi }$ denote operations with rotation angles $\theta$ about the $x/y/\varphi$ axes that are resonant with the transition $0\leftrightarrow -1$ and with Rabi frequencies of 8 MHz. (a) Pulse sequence to demonstrate the effect of $U_{\mathrm{CNOT}}$ on different input states via electron spin detection. $\varphi$ is the detuning phase. In the presence (absence) of the $180_y$ operation indicated by the dashed box, the FID measurement is used to determine the population of the $m_S=-1$ ($m_S=0$) after $U_{\mathrm{CNOT}}$. (b) Pulse sequence to demonstrate the effect of $U_{\mathrm{CNOT}}$ via ${}^{13}\mathrm{C}$ spin detection. (c) Pictorial representation of state ${\psi }_1$.

Figure 4

(a) Electron spin spectra for the pulse sequence corresponding to Fig. 3 without and with $U_{\mathrm{CNOT}}$ where ${\theta }_y=\pi$. The thermal state spectra on top are shifted vertically for reference. The electron spin spectra are centered around the detuning frequency 3 MHz. (b) ${}^{13}\mathrm{C}$ spin spectra obtained by the pulse sequence shown in Fig. 3 without and with $U_{\mathrm{CNOT}}$. The peaks appear at ${\nu }_{-}=0.11\mathrm{MHz}$.

Figure 5

$P_{0\downarrow }$ as a function of $\theta$ corresponding to the pulse sequences shown in Fig. 3. The diamonds and solid circles are the experimental data, and the dashed lines are the matching simulations.