Experimental Realization of Robust Geometric Quantum Gates with Solid-State Spins

We experimentally realize a universal set of single-bit and two-bit geometric quantum gates by adiabatically controlling solid-state spins in a diamond defect. Compared with the nonadiabatic approach, the adiabatic scheme for geometric quantum computation offers a unique advantage of inherent robustness to parameter variations, which is explicitly demonstrated in our experiment by showing that the single-bit gates remain unchanged when the driving field amplitude varies by a factor of 2 or the detuning fluctuates in a range comparable to the inverse of the gate time. The reported adiabatic control technique and its convenient implementation offer a paradigm for achieving quantum computation through robust geometric quantum gates, which is important for quantum information systems with parameter-fluctuation noise such as those from the inhomogeneous coupling or the spectral diffusion.

Figure 1

Relevant energy levels and microwave shape schemes for the adiabatic geometric single-qubit gates in a diamond NV center. (a) The relevant energy level structure of the electron spin in an NV center under a magnetic field. The two levels encoding a qubit are coupled by a microwave pulse with the Rabi frequency $\mathrm{\Omega }$ and the detuning $\mathrm{\Delta }$. (b) The microwave pulse shape used to achieve a geometric single-qubit rotation operation along the $x$ axis. $\mathrm{\Omega }(t)$ is in the form of $\mathrm{\Omega }(t)/{\mathrm{\Omega }}_m=1-|\sin (2\pi t/T){|}^n$ [26] with ${\mathrm{\Omega }}_m$, $T$, and $n$ being the maximum value of $|\mathrm{\Omega }|$, the gate time, and a positive integer, respectively. We take $n=5$ for our experiment. $\mathrm{\Delta }(t)$ consists of piecewise linear functions of time with the sign being suddenly reversed at $t=T/4$ and $3T/4$ and ${\mathrm{\Delta }}_m$ is its maximal value. Phase shifts $\mathrm{\Delta }{\varphi }_1=\pi +\varphi /2$ and $\mathrm{\Delta }{\varphi }_2=-\pi -\varphi /2$ are inserted into the microwave at $t=T/4$ and $3T/4$, respectively. (c) The microwave pulse shape used to realize a rotation operation about the $z$ axis. A phase $\mathrm{\Delta }\varphi =\pi +\varphi /2$ is inserted into the microwave at $t=T/2$. (d) Evolution of a state initialized to an instantaneous eigenstate of the Hamiltonian (4) in the Bloch sphere, showing the geometric phase which equals half of the enclosed solid angle $\varphi$. The green and purple arrows denote the initial eigenstate respectively for scheme shown in (b),(c).

Figure 2

Measured fidelity for the adiabatic geometric singe-qubit gates. (a) Average fidelity as a function of the number of gates. Orange circles, green diamonds, and purple triangles denote the results from a standard randomized benchmarking (RB) protocol, a interleaved $Z_{\pi /2}$ and a interleaved $X_{\pi /2}$ gate, respectively. Each point in the figure is obtained by averaging the experimental data from 20 measurements. These points are fitted by $F=A{p}^m+B$ plotted as solid lines with the corresponding colors. (b) Fidelity of the final states measured by the quantum state tomography for $Z_{\pi /2}$, $Z_{\pi /8}$, and $X_{\pi /2}$ gates acting on six distinct initial states. For each gate, the parameters of microwave pulses are $T=1\mu s$, ${\mathrm{\Omega }}_m=20\mathrm{MHz}$, and ${\mathrm{\Delta }}_m=20\mathrm{MHz}$. Note that the number in the bracket following the fidelity value represents the error bar (s.d.) in the last decimal place.

Figure 3

Experimental results demonstrating robustness of adiabatic geometric single-qubit gates with respect to variations of ${\mathrm{\Omega }}_m$ (the random coupling) and $\delta$ (the spectral diffusion). The detuning is modeled by $\mathrm{\Delta }(t)+\delta$. Normalized photon luminance as a function of the imprinted phase $\varphi$ under three distinct values of $\delta$ and ${\mathrm{\Omega }}_m$, respectively, for the rotation operation along the $x$ axis (a),(b) and the rotation operation along the $z$ axis (c),(d). The other parameters are ${\mathrm{\Delta }}_m=20\mathrm{MHz}$, $T=1\mu \mathrm{s}$, and ${\mathrm{\Omega }}_m=20\mathrm{MHz}$ in (a),(c), while in (b),(d) ${\mathrm{\Delta }}_m=20\mathrm{MHz}$, $T=1\mu \mathrm{s}$, and $\delta =0\mathrm{MHz}$. Note that for the $Z$ rotation gate, two half $\pi$ pulses around the $x$ axis are applied in front and after the target gate in order to transfer the phase information to the photon luminance count.

Figure 4

Level scheme, pulse sequence, and experimental results for the geometric two-qubit crot gate. (a) Energy level structure of the electron and nuclear spins for the geometric crot gate together with microwave and radio-frequency (rf) coupling configuration. (b) Time sequence for implementing and detecting the crot gate. We first use a MW0 $\pi$ pulse followed by illumination of a 532 nm green laser pulse for $2\mu \mathrm{s}$ to initialize the system and then apply MW1, MW2, and rf pulses to create a desired state. The spin echo for implementation of the geometric crot gate is realized by simultaneously applying two $\pi$ pulses denoted as MW1 and MW2. Other parameters for the crot gate are $n=5$, $T=2\mu s$, ${\mathrm{\Delta }}_m=7\mathrm{MHz}$, and ${\mathrm{\Omega }}_m=4\mathrm{MHz}$. Two-qubit quantum state tomography is used to measure the fidelity of the final state. To avoid the decoherence of the electron spin during the slow RF pulse, a spin echo of 1 MHz is inserted in the middle of the quantum state tomography process. (c) Measured fidelity of the final states after applying the crot gate to six complementary initial states. (d) Measured real and imaginary parts of the final state matrix elements for the geometric crot gate applied to the initial state $|1⟩(|\uparrow ⟩+|\downarrow ⟩)$ compared with the matrix elements under the ideal gate represented by the hollow caps.