Experimental Implementation of Universal Holonomic Quantum Computation on Solid-State Spins with Optimal Control

Experimental realization of a universal set of quantum logic gates with high fidelity is critical to quantum-information processing, which is always challenging due to the inevitable interaction between the quantum system and environment. Geometric quantum computation is noise immune, and thus, offers a robust way to enhance control fidelity. Here, we experimentally implement the recently proposed extensible nonadiabatic holonomic quantum computation with solid spins in diamond at room temperature, which maintains both flexibility and resilience against decoherence and system-control errors. Compared with the previous geometric method, the fidelities of a universal set of holonomic single-qubit and two-qubit quantum logic gates are improved in experiments. Therefore, this work makes a step towards high-fidelity quantum-information processing based on extensible nonadiabatic holonomic quantum computation in realistic systems.

Figure 1

(a) Schematic of the unit cell of diamond, including an $\mathrm{N}$-$V$ center. (b) Encoding of a qubit in the spin-triplet ground state and microwave coupling configuration. (c) Conceptual explanation of geometric quantum operation in the Bloch sphere, showing the geometric phase that equals half of the enclosed solid angle, ${\gamma }_G$. (d) Schematic of different paths for the NHQC (red) and NHQC+ (brown) with optimization methods. (e) Holonomic $X$ gate on a bright state. Experimental results (dots) fit well with simulations (solid lines) with $\eta =1$. (f) Performance of an $X$ gate with control errors ($\alpha$) for NHQC and NHQC+ schemes with initial state $|0⟩$. Envelopes of the two driving fields of NHQC are truncated Gaussian pulses [18, 19].

Figure 2

QPT results of single-qubit operation. (a) Experimental pulse sequence to characterize the performance of the single-qubit gate. (b) Bar charts of the real and imaginary parts of ${\chi }_{\exp }$ for specific gates $I$, $X/2$, $Y/2$, and $X$, giving fidelities of $0.983(4)$, $0.975(5)$, $0.978(5)$, and $0.977(5)$, respectively. Labels on the $X$ and $Y$ axes correspond to operators in the basis set $\left\{I,{\sigma }_x,{\sigma }_y,{\sigma }_z\right\}$ of the $\{|0⟩,|1⟩\}$ subspace. $X/2$ ($Y/2$) is a $\pi /2$ rotation around the $x$ ($y$) axis in the Bloch sphere.

Figure 3

Decay of the fidelity of nonadiabatic holonomic $X/2$ and $Y/2$ gates obtained via QPT. Experimental data for $X/2$ (red dots) and $Y/2$ (blue dots) gates are fitted by $F=\left[1+\left(1-{\epsilon }_{\text{IF}}\right){\left(1-2p\right)}^N\right]/2$ (solid line), and ${\epsilon }_{\text{IF}}$ describes errors in state preparation and measurement. Gold line represents the necessary fidelity threshold of $99\mathrm{\%}$ for the implementation of state-of-the-art error-correction codes based on surface codes.

Figure 4

(a) Level structures of electron and nuclear spins for the geometric crot gate and the MW and rf coupling configuration. (b) Time sequence for the implementation and verification of the geometric crot gate between electron and nuclear spins. crot gate is implemented by applying rf1 and rf2 pulses simultaneously. Duration time for crot gate is $42\mu \text{s}$. MW pulses are used to implement a spin echo to increase the $\mathrm{N}$-$V$ center’s coherence time, and the selective MW sequence is ${\pi }_{\text{MW3}}-{\pi }_{\text{MW2}}-{\pi }_{\text{MW1}}-{\pi }_{\text{MW4}}$. To verify the crot gate, we use a combination of MW1–MW4 and a rf to implement QST. (c) Matrix elements of the output density operator reconstructed through QST, where geometric crot is applied to the product state $|0⟩-|1⟩/\sqrt{2}|1⟩$.

Figure 5

(a),(b) Amplitude and phase of MW field used in experiments for the NHQC+ $X$ gate. (c) Waveform pulse of $4\sigma$-truncated Gaussian shape in the conventional NHQC $X$ gate without phase modulation. (d) Dynamical phases of the $X$ gate in NHQC+ (orange curve) and NHQC (red dashed line) schemes as a function of time $t$ during the full evolution.

Figure 6

(a) Amplitude of MW field in experiments for single-qubit holonomic gates. (b),(c) Phases of MW field used in experiments for NHQC+ $X/2$ and $X$ gates with $\eta =0.4$. For the $Y/2$ gate, $\varphi =\pi /2$ and the amplitude- and phase-modulation sequence are the same as those of the $X/2$ gate.