High-fidelity control of a nitrogen-vacancy-center spin qubit at room temperature using the sinusoidally modulated, always rotating, and tailored protocol

A practical implementation of a quantum computer requires robust qubits that are protected against their noisy environment. Dynamical decoupling techniques have been successfully used in the past to offer protected high-fidelity gate operations in negatively charged nitrogen-vacancy (${\mathrm{NV}}^{-}$) centers in diamond, albeit under specific conditions with the intrinsic nitrogen nuclear spin initialized. In this work, we show how the sinusoidally modulated, always rotating, and tailored (SMART) protocol, an extension of the dressed-qubit concept, can be implemented for continuous protection to offer Clifford gate fidelities compatible with fault-tolerant schemes, whilst prolonging the coherence time of a single ${\mathrm{NV}}^{-}$ qubit at room temperature. We show an improvement in the average Clifford gate fidelity from $0.940\pm 0.005$ for the bare qubit to $0.993\pm 0.002$ for the SMART qubit, with the nitrogen nuclear spin in a random orientation. We further show a $\gtrsim 30$ times improvement in the qubit $T_2^{*}$ coherence times compared to the bare qubit.

Figure 1

Single-qubit ODMR. (a) An optical second-order correlation measurement showing a normalized $g^{\left(2\right)}\left(0\right)$ value of 0.18 confirms photon emission from a single ${\text{NV}}^{-}\text{center}$. (b) Hyperfine levels within the $|0\rangle \leftrightarrow |+1\rangle$ MW transition manifold showing a separation of $\sim 3.1\mathrm{MHz}$ consistent with literature for ${}^{15}\mathrm{NV}$-based ${\text{NV}}^{-}$ defects. (c) Coherent Rabi oscillations observed when driving the $|0\rangle \leftrightarrow |+1\rangle$ electronic transition with the MW frequency equally detuned from both hyperfine levels. (d) Ramsey oscillations observed showing the free-induction decay of the electronic spin at roughly half the hyperfine splitting.

Figure 2

A SMART Ramsey measurement. (a) The SMART Ramsey pulse sequence. The $\pm X/2$ pulses and always-on field are applied through the I channel of the MW source. (b) Measured coherence times as a function of modulation period $T_{\mathrm{mod}}$. The filled area represents the bounds of the error bars. We find that the transverse coherence time exceeds $\sim 30\mu \mathrm{s}$ at optimum modulation periods indicated in the plot. This represents at least a factor of 30 improvement in the $T_2^{*\mathrm{SMART}}$ coherence time compared to the bare spin. The normalized $T_2^{*\mathrm{SMART}}$ data is fitted to an absolute valued Bessel function represented by a solid line, showing that the optimum $T_{\mathrm{mod}}$ periods are found at the $j_i\mathrm{th}$ solution of the Bessel function as summarized in Table 2.

Figure 3

Control signals to implement two-axis control for the dressed and SMART qubits. Rabi oscillations indicate coherent $|-\rangle \leftrightarrow |+\rangle$ transitions. (a) We implement frequency modulated control signals, with the FM signal amplitude modulated at a frequency equal to the bare qubit Rabi frequency. The resulting dressed Rabi frequency is then tuned by adjusting the FM depth. (b) To implement two-axis control on the SMART qubit we use amplitude modulation on the IQ MW channels at frequencies $f_{\mathrm{mod}}$ and $2f_{\mathrm{mod}}$, where the qubit coherence is optimized by using $T_{\mathrm{mod}}=T_{\mathrm{opt}}$ from Table 2.

Figure 4

Rabi oscillations in the dressed basis similar to Fig. 3 for longer drive times. Compared to the bare spin, the $T_2^{\mathrm{Rabi}}$ coherence time in the dressed basis may be prolonged by a factor of $\sim 3$ using simple qubit dressing, and by a factor of $\sim 8$ using the SMART protocol, consistent with coherence improvement shown in Fig. 2. (a) Rabi oscillations of the dressed qubit. (b) Rabi oscillations of the SMART qubit.

Figure 5

Randomized benchmarking of the bare, dressed, and SMART qubits. All measurements are based on ${\mathrm{\Omega }}_{\mathrm{R}}=9\mathrm{MHz}$ on the bare spin. The average Clifford gate fidelity $F_{\mathrm{C}}$ is extracted using the relevant fitting function (see text) with 95% confidence interval. The error bars of individual data points are dominated by signal averaging. Additionally, we calculate the theoretical UB from the measured Rabi quality factors. The y-intercept for the randomized benchmarking signals is determined by state preparation and readout error. (a) Measurement of $F_{\mathrm{C}}$ for the bare spin qubit is in good agreement with the simulation, based on the decay rate as a function of Clifford gates applied and the extracted average Clifford gate fidelity. This close agreement is in absence of control signal noise or detuning noise in the simulation, suggesting that the primary source of low fidelity is the $\sim 1.5\text{−}\mathrm{MHz}$ detuning caused by the fluctuating nuclear spin. (b) $F_{\mathrm{C}}$ for the dressed and SMART spin qubit protocols, measured for primitive gates defined by 0.95 MHz dressed Rabi frequency and $1/4T_{\mathrm{mod}}=0.96$ MHz SMART Rabi frequency, respectively. The average Clifford gate fidelities for both protocols are expected to be limited by the spin coherence time. We observe $>0.99$ average Clifford gate fidelity for approximately the same Rabi frequency as for the dressed qubit ($\sim 0.95\mathrm{MHz}$), demonstrating a significant improvement consistent with enhanced Rabi coherence times.