Parallel selective nuclear-spin addressing for fast high-fidelity quantum gates

Due to their long coherence times, nuclear spins have gained considerable attention as physical qubits. Two-qubit gates between nuclear spins of distinct resonance frequencies can be mediated by electron spins, usually employing a sequence of electron-nuclear gates. Here we present a different approach inspired by, but not limited to, NV centers in diamond and discuss possible applications. To this end we generalize external electron spin control sequences for nuclear spin initialization and hyperpolarization to achieve the simultaneous control of distinct nuclear spins via an electron spin. This approach results in efficient entangling gates that, compared to standard techniques, reduce the gate time by more than 50% when the gate time is limited by off-resonant coupling to other spins and by up to 22% when the gate time is limited by small electron-nuclear coupling.

Figure 1

Two basic robust tunable pulsed polarization sequences are shown on the left top with pulse duration on the top of each pulse in radians and pulse phase at the bottom in degrees. (a) PulsePol [9]; (b) XY like: The parameter $\varphi$ allows to manipulate the resonances as explained in the main text, the numbers in the pulses indicate pulse length and phase, the sequence is repeated $2N$ times. (c) The circle shows a visualization of the intuitive picture described in the text: To be resonant, the nuclear spin must cover a total angle of $\pi +\varphi$ (green dashed arrow). Rotating in the opposite direction also achieves resonance (red solid arrow), as well as adding multiples of $2\pi$ (blue dotted). These resonances are at their expected position as illustrated in panel (d), where the black line indicates the XY resonance and the others correspond to the arrows in the right figure. If on resonance with the nuclear spin, then the effective Hamiltonian transfers polarization to it, but the direction [flipflip or flipflop Hamiltonian $H\sim {\sigma }_x{I}_x\pm {\sigma }_y{I}_y$, corresponding to clock- or anticlockwise rotation in panel (c)] depends on the resonance.

Figure 2

(a) Sequence used for all following simulations: The $\pi$ pulses in the sequence of Fig. 1 (a) are replaced by 5 pi pulses, introducing the parameters ${\tau }_{0,1,2}$. (b) Process infidelity for different detuning and amplitude errors using the (0,$+$) and (1,$-$) conditions; for details see text. (c) Schematic visualization of the situation considered: An NV center is used to entangle two nuclear spins of frequencies ${\omega }_{1/2}$, while a third spin of frequency ${\omega }_3$ inhibits the gate. (d) Infidelity as a function of the allowed total gate time and the frequency of an additional spin.

Figure 3

Process infidelity in the relevant nuclear subspace {$|10\rangle ,|01\rangle$} of a gate that evolves a state $|1\rangle \otimes |10\rangle \otimes |+\rangle$ to $|1\rangle \otimes \left(|10\rangle +|01\rangle \right)/\sqrt{2}\otimes |+\rangle$ while being disturbed by a third spin like in Fig. 2. Here the conditions (22,$-$) and (23,$-$) were used, as a result the maximally achievable coupling strength is reduced and therefore the gate times are longer.

Figure 4

Comparison of process infidelities of the direct gate approach (present work) and the sequential gate approach [13]. For the parameters of Fig. 2 our proposed direct gate approach is shown for the fastest possible time $T\cong 120 \mu \mathrm{s}$ (black dash-dotted line) and $T\cong 140 \mu \mathrm{s}$ (red solid line), which is the fastest possible gate time achievable with a sequential gate approach. The latter is simulated using the ideal case of instantaneous-pulse CPMG sequences (blue dashed line and its envelope in green dotted line). The regions where a third nucleus affects the gate are smaller for the polarization-based sequences. The FWHM for equal total gate times that characterizes the range of frequencies that impair the gate fidelity is almost three times worse for the electron-nuclear gate envelope compared to the polarization sequence approach. A factor 2 that stems from the fact that electron-nuclear gates only use half of the sequence time to decouple the individual nuclei can be seen in the blue curve (FWHM), its sinc shape is the reason for the bigger envelope, see text for discussion.

Figure 5

The modulation function $f_1$ for the adapted sequence until the center of the central $\pi$ pulse ($t=\tau$) rises to 1 during first $\pi$/2 pulse and remains there until the beginning of the composite pulse. For symmetry reasons it is sufficient to consider only the first $\pi$ pulse ($1\to -1$), the evolution afterwards that takes a fraction ${\beta }_1$, the second $\pi$ pulse ($-1\to 1$), the evolution afterwards that takes a fraction ${\beta }_2$ and half of the third $\pi$ pulse ($1\to 0$).

Figure 6

Comparison of effective coupling (left) for $A_{\text{eff}}({\varphi }_1,{\beta }_1,{\beta }_2)$ (top) and $A_{\text{eff}}(2\pi -{\varphi }_1,{\beta }_1,{\beta }_2)$ (bottom) for the parameters in Fig. 2 (main text) and difference between those functions (right). Here ${\tau }_{1,2}={\beta }_{1,2}\frac{\pi +\varphi }{2{\omega }_L}$ was used as dimensionless time ${\beta }_i<1$. The chosen ${\beta }_{1,2}$ is marked with a white cross where bot functions were equal and close to a desired value.