Implementation of Dynamically Corrected Gates on a Single Electron Spin in Diamond

Precise control of an open quantum system is critical to quantum information processing but is challenging due to inevitable interactions between the quantum system and the environment. We demonstrated experimentally a type of dynamically corrected gates using only bounded-strength pulses on the nitrogen-vacancy centers in diamond. The infidelity of quantum gates caused by a nuclear-spin bath is reduced from being the second order to the sixth order of the noise-to-control-field ratio, which offers greater efficiency in reducing infidelity. The quantum gates have been protected to the limit essentially set by the spin-lattice relaxation time $T_1$. Our work marks an important step towards fault-tolerant quantum computation in realistic systems.

Figure 1

(a) A spin qubit interacting with the nuclear-spin bath. (b) Schematic of the optically detected magnetic resonance setup. The external magnetic field $B_0$ was applied by a permanent magnet, while the microwave pulse was carried by a waveguide. (c) Energy-level diagram of the NV center with external magnetic field $B_0$ (upper panel) and a continuous-wave spectrum of $|0⟩\leftrightarrow |1⟩$ with $B_0=513\mathrm{G}$ (lower panel). (d) FID of the electron spin. Experimental data (orange line) are fitted by $y=0.5-0.5\exp [-{(t/T_2^{*})}^2]\cos (2\pi \omega t)$ (blue line) with $\omega =0.5\mathrm{MHz}$ and $T_2^{*}=6.56(0.17)\mu \mathrm{s}$.

Figure 2

(a) Pulse sequences corresponding to a plain sequence and to three- and five-piece supcode sequences. The calculation of time durations ${\tau }_i$ ($i=1$, 2, 3, and 4) is included in the Supplemental Material [22] with ${\tau }_0=1/{\omega }_1$ and ${\omega }_1=20=\mathrm{MHz}$. (b) The measured and theoretically predicted state infidelities of the destination states as functions of $\delta /{\omega }_1$. Experimental data fit the theoretical predictions (lines) well. Some deviations are caused by statistical fluctuations of photon numbers, with the uncertainty plotted as a dashed gray line.

Figure 3

Universal control of a single spin by the five-piece supcode pulse. The left panels of (a) and (b) show the Bloch-sphere diagrams of the spin vectors rotating around (a) $\mathbf{x}$ and (b) $\mathbf{y}$, respectively. The right panels are the corresponding detected coherent oscillations of the qubit driven by the five-piece SUPCODE pulses. Light red (blue) lines are the theoretical predictions with $X$ ($Y$) detection, and red (blue) rectangles are the corresponding experimental data.

Figure 4

(a) Comparison of the quantum oscillation driven by five-piece supcode pulses with the one driven by plain pulses. Black crosses are the supcode experimental data, and red lines are the envelopes of the function $0.5+0.5\exp (-T/T_{\mathrm{DCG}})\cos (\omega T)$ (light gray line), with $T_{\mathrm{DCG}}=690(40)\mu \mathrm{s}$. The measured (blue diamonds) quantum oscillation by the plain pulses (Rabi oscillation) was fitted by $0.5+0.5\exp (-T/T_2^{\prime })\cos ({\omega }^{\prime }T)$ (blue lines) with $T_2^{\prime }=135(10)\mu \mathrm{s}$. (b) The decay of the fidelity of five-piece supcodes obtained via quantum process tomography. The black line is the theoretically predicted fidelity when only the $T_{1\rho }$ process is taken into account. The shaded region is calculated with the uncertainty of the measured $T_{1\rho }$. Blue triangles are experimental fidelities by QPT, which shows a $T_{1\rho }$-limited decay. (c) Extended coherence times via CPMGs. $N$ stands for the number of the $\pi$ pulses. After applying more than 1000 pulses, the prolonged coherence times are limited by $T_{1\rho }=660(80)\mu \mathrm{s}$. (d) Comparison of the decay times.