High Fidelity Control of a Nitrogen-Vacancy Spin Qubit at Room Temperature using the SMART Protocol

The negatively charged Nitrogen-Vacancy (NV^-) centre in diamond [1] has long been proposed as a potential qubit platform for room-temperature quantum computing, owing to its long coherence times and compatibility with optically detected magnetic resonance techniques [2, 3, 4]. The longitudinal relaxation time $T_{1}$ and transverse relaxation time $T_{2}$, together dictate how long a spin-qubit can remain in a prepared state. These metrics of coherence are of general importance for applications in quantum computing and memory nodes for quantum communication [5, 6, 7].

The overall coherence of an NV^-  spin qubit is typically limited by a relatively shorter $T_{2}$ coherence time, due to decoherence produced by a nuclear spin bath [8] in isotopically unpurified diamond. The nuclear spin bath, along with the host nuclear spin of the NV^- , therefore negatively impact qubit gate operations, leading to lower gate fidelities [9].

Qubit gate operations with fidelities exceeding 0.99 are said to allow for error-correction strategies which are critical for qubit scalability [10, 11, 12]. To achieve this, dynamical decoupling techniques may be employed to protect the qubit [13, 9, 14, 15]. In addition, universal holonomic gates [16, 17] and composite pulse sequences [18] have also been demonstrated to exhibit high gate fidelities. However, in most cases a static magnetic field (B₀) aligned with the NV^- main symmetry axis is required to polarise the host nitrogen nuclear spin [19, 20], which in turn reduces the occurrence of detuned gate operations. While helpful for gate fidelities, initialisation of the nuclear spin presents an added layer of complexity towards scalability, requiring either RF delivery or careful alignment of the B₀ field for multiple NV^- spin qubits. If the host nuclear spin is of no interest, it will be useful to perform high-fidelity qubit gate operations on the electronic spin without nuclear spin initialisation.

Alternative control strategies include the implementation of microwave (MW) dressed states, which protect the qubit against magnetic field fluctuations by offering continuous dynamical decoupling [21, 22, 23, 24]. The always-on MW field redefines the quantization axis along the driving axis, and the quantization energy according to the driving strength. The latter opens up alternative two-axis control methods [25, 26]. Recently, the Sinusoidally Modulated, Always Rotating and Tailored (SMART) protocol as an extension to the dressed protocol, was employed to achieve high fidelity gates in a SiMOS quantum dot device at low temperatures [27, 28]. In this protocol the resonant MW drive is sinusoidally modulated to reduce the impact of MW detuning and amplitude fluctuations; in particular, the effect of detuned driving has a significant impact on idle qubits (identity gate operation). Some advantages of the SMART protocol over other methods, such as dynamical decoupling sequences, include simplicity in implementation, effective protection against decoherence even during gate operations and compatibility with global control of spin qubits [29, 30]. Similar alternating MW fields have also recently been employed to protect qubits in hBN from a nuclear spin bath [31], and to optically address NV^- spins using universal holonomic gates in diamond [17] at room temperature.

In this work, we utilize the SMART protocol to demonstrate room temperature, high-fidelity, single qubit gates on the negatively charged Nitrogen-Vacancy (¹⁵NV^-) electron spin, hosted within an electronic grade CVD diamond sample. We first measure the bare qubit coherence times and compare them to the extended coherence time using the SMART protocol. We then construct various single qubit gates in the bare, dressed, and SMART qubit bases and extract the average Clifford gate fidelities using randomized-benchmarking techniques [32]– thereby demonstrating high control fidelities compatible with fault-tolerant architectures and the advantage of the SMART protocol.

We first implement a bare spin qubit using the $\ket{0}\leftrightarrow\ket{+1}$ spin transition of a single NV^- electron spin with a ¹⁵N isotope nuclear spin ($I=1/2$). The chemical vapour deposition grown diamond sample used in this work contains etched nanopillars for improved photon collection efficiency and better isolation of single NV^- defects with a non-resonant laser [33]. The Hamiltonian of the NV^- spin is written in the laboratory frame as

where $D$ is the zero-field splitting of $\sim 2.87$ GHz, {S, I} are operators for the electron and nuclear spins respectively, B₀ is the static magnetic field, {$\gamma_{\rm e}$, $\gamma_{\rm n}$} are gyromagnetic ratios for the electron and ¹⁵N nuclear spins respectively, and {$A_{\rm\perp}$, $A_{\rm||}$} are hyperfine constants. The qubit spin state is correlated with the photoluminescence intensity of the NV^- electronic spin, and is measured using suitable optically detected magnetic resonance (ODMR) techniques on a measurement setup similar to previous work [34, 35]. A $g^{(2)}(\tau)$ measurement confirms a single NV^- spin with $g^{(2)}(0)$ $\ll 0.5$ as shown in Fig. 1(a). An ODMR sweep measurement shown in Fig. 1(b) reveals that the detected NV^- centre is hyperfine coupled to a ¹⁵N host nuclear spin with a strength of $\sim$ 3.1 MHz, consistent with literature [36, 37].

We perform randomized benchmarking on the bare, dressed and SMART qubits to compare their overall average Clifford gate fidelity $F_{\rm C}$, extracted from the fit function $P(N)=P_{0}(2F_{\rm C}-1)^{N}+P_{\infty}$ [32]. Here $N$ is the number of Clifford gates, $P(N)$ is the spin probability as a function of applied Clifford gates, and variables $P_{\infty}$, $P_{0}$ absorb the state preparation and measurement error [39]. Importantly the variable $P_{\infty}$ is fixed to an experimentally determined value, while $P_{0}$ and $N$ are left as free parameters for the fit. Additionally, we calculate the theoretical upper bound (UB) for $F_{\rm C}$ using $1-1.875/(4Q_{\rm R})$[40, 32], where $Q_{\rm R}$ is the measured Rabi quality factor.

We also simulate randomized benchmarking on the bare qubit using time-domain simulation of the NV^- Hamiltonian shown in Eq. 1. For the measurements we average over the complete gate space constructed from $\rm\{X,\pm{X/2},Y,\pm{Y/2}\}$ primitive gates for $k\geq 30$ repetitions, while for the simulations we average for $k=50$ repetitions to ensure convergence. Additionally, we repeat the measurements for both target output states $\ket{+1},\ket{0}$ in the bare spin basis and $\ket{+},\ket{-}$ in the dressed spin basis, and combine the results to yield an average Clifford gate fidelity. The average Clifford gate fidelities for the bare, dressed and SMART qubits are compared in Fig. 5.

As seen in Fig. 5(a), a simulation of the randomized benchmarking for the bare qubit without decoherence modelling(black symbols) is able to accurately predict the measured average Clifford gate fidelity of $F_{\rm C}^{\rm bare}=0.940\pm 0.005$ (red symbols). The upper bound for the bare $F_{\rm C}$ calculation is higher than this value ($<$ 0.995) since $Q_{\rm R}^{\rm bare}$ is not affected by detuned driving. These results suggest that the main limiting factor for the bare qubit control fidelity is the detuning of 1.5 MHz due to the uninitialised nuclear spin. The dressed qubit with a 0.95 MHz Rabi frequency however offers a much better gate fidelity of $F_{\rm C}^{\rm dressed}=0.978\pm 0.002$; the main limiting factor for the dressed qubit is attributed to spin decoherence. Higher dressed Rabi frequencies may be problematic due to breakdown of the rotating-wave approximation, unless the bare Rabi frequency is also increased [38]. For practical difficulties in increasing the MW power, we are limited to the bare Rabi frequency of 9 MHz.

For the SMART qubit, the gate lengths are restricted to a multiple of $T_{\rm opt}$. Thus, with $T_{\rm opt}=260$ ns, we calibrate the SMART Rabi frequency to $1/(4T_{\rm opt})$. The SMART protocol results in even higher average Clifford gate fidelity of $F_{\rm C}^{\rm SMART}=0.993\pm 0.002$, exceeding 0.99, for a similar Rabi frequency of $\sim{}0.96$ MHz, and achieving results similar to [28]. This is expected as the SMART qubit exhibits an extended coherence time, as can be seen from inspection of Fig. 2 and Fig. 4. Although Fig. 2 shows a marginally better coherence improvement for the second $T_{\rm opt}$ operating setpoint of 588 ns, we expect that the reduction in the resulting Rabi frequency would not be of benefit for achieving higher gate fidelities. Future work in this direction may explore using multi-tone amplitude modulation instead to work around this limitation [28].

High fidelity control of qubits with gate fidelities exceeding 0.99 is critical for implementing various fault-tolerant quantum computing circuits. In this work we have shown that the SMART protocol enables high fidelity control of the electronic spin of an NV^-  spin qubit at room temperature, without needing to initialise the hyperfine-coupled host nuclear spin. We first demonstrate a factor of 30 improvement in the $T_{2}$ coherence time of the SMART qubit when compared to the bare qubit. We then measure an average Clifford gate fidelity of 0.993$\pm$0.002 for the SMART qubit using randomized benchmarking techniques. We find that the SMART protocol achieves gate fidelities comparable to dynamical decoupling methods albeit with a simpler and environment-agnostic implementation, thereby simplifying experimental requirements and qubit operations, and providing a scalable path for multi-qubit operation.

We acknowledge support from the Australian Research Council (CE170100012). H.H.V. and I.H. acknowledge support from the Sydney Quantum Academy. H.H.V also acknowledges support from the Australian Government Research Training Program Scholarship. C.A. and A.L. acknowledge support from the University of New South Wales Scientia program.