Protected Quantum Computing: Interleaving Gate Operations with Dynamical Decoupling Sequences

Implementing precise operations on quantum systems is one of the biggest challenges for building quantum devices in a noisy environment. Dynamical decoupling attenuates the destructive effect of the environmental noise, but so far, it has been used primarily in the context of quantum memories. Here, we experimentally demonstrate a general scheme for combining dynamical decoupling with quantum logical gate operations using the example of an electron-spin qubit of a single nitrogen-vacancy center in diamond. We achieve process fidelities $>98\%$ for gate times that are 2 orders of magnitude longer than the unprotected dephasing time $T_2$.

Figure 1

Single cycle of an $XY-4$ DD sequence used as the basis of protected gate operations. The labels $x$ and $y$ mark the rotation axes of the DD pulses, and ${\mathcal{H}}_n$ is the Hamiltonian between the DD pulses.

Figure 2

Laser (top) and microwave (bottom) pulse sequences for the not gate protected by an $XY-4$ DD cycle. The duration of the laser pulses is 400 ns. The pulse sequence (duration ${\tau }_c$) between the first and last pulses implements the protected not gate, where unfilled boxes represent the DD pulses and filled ones represent the segments of the gate operation.

Figure 3

$\chi$ matrices measured by quantum process tomography for the noop, not, Hadamard, and phase gates protected by $XY-8$ dynamical decoupling pulses, for gate durations of $\approx 35.5\mu \mathrm{s}$. The first row shows the real parts of the noop, not, and Hadamard gates, and the second row shows real and imaginary parts of the phase gate. The imaginary parts for the noop, not, and Hadamard gates are not shown; their rms values are $<0.03$, which is compatible with zero within experimental uncertainties.

Figure 4

The measured gate fidelity obtained by quantum process tomography for the gates without and with dynamical decoupling pulses. The curves are functions $A{e}^{-(t/T_2{)}^k}$, with the fit parameters of Table III in the SM.

Figure 5

Effect of the hyperfine coupling between the electron and the ${}^{14}\mathrm{N}$ nuclear spin on the fidelity of the Hadamard gate. The measured fidelity of the gate protected by an $XY-8$ cycle is shown as the solid thick curve and the simulated fidelity for the same gate as the dashed thick curve. The dashed thin curve shows the fidelity of the unprotected gate by simulation and the solid thin curve the corresponding experimental data. The inset shows how the maximal loss of the gate fidelity decreases with increasing Rabi frequency of the control pulses according to the simulation. The point on the solid thick curve, at a Rabi frequency of 12.7 MHz, indicates the value corresponding to the main plot.