Single-Loop Realization of Arbitrary Nonadiabatic Holonomic Single-Qubit Quantum Gates in a Superconducting Circuit

Geometric phases are noise resilient, and thus provide a robust way towards high-fidelity quantum manipulation. Here we experimentally demonstrate arbitrary nonadiabatic holonomic single-qubit quantum gates for both a superconducting transmon qubit and a microwave cavity in a single-loop way. In both cases, an auxiliary state is utilized, and two resonant microwave drives are simultaneously applied with well-controlled but varying amplitudes and phases for the arbitrariness of the gate. The resulting gates on the transmon qubit achieve a fidelity of 0.996 characterized by randomized benchmarking and the ones on the cavity show an averaged fidelity of 0.978 based on a full quantum process tomography. In principle, a nontrivial two-qubit holonomic gate between the qubit and the cavity can also be realized based on our presented experimental scheme. Our experiment thus paves the way towards practical nonadiabatic holonomic quantum manipulation with both qubits and cavities in a superconducting circuit.

Figure 1

Single-loop single-qubit holonomic gates. (a) Two microwave fields are resonantly coupled to the $|g⟩\leftrightarrow |e⟩$ and $|e⟩\leftrightarrow |f⟩$ transitions of a transmon qubit to generate arbitrary single-qubit holonomic gates in the $\{|g⟩,|f⟩\}$ subspace. The first excited state $|e⟩$ is an auxiliary level. (b) Bloch sphere representation of the holonomic gate: a combination of two microwave fields, with two different sets of phases for the first and the second half of the gate operation, equivalently drives the bright state $|b⟩$ to the auxiliary state $|e⟩$ and then back with an additional phase, while the dark state $|d⟩$ remains unchanged. (c) Simplified experimental setup. Two transmon qubits in two microwave trenches are coupled to two microwave cavities, one for readout and the other one for storage. The qubit holonomic gates are demonstrated with $Q_1$, while the cavity holonomic gates are realized with the Fock states $\{|0⟩,|1⟩\}$ in the storage cavity facilitated by both qubits (Fig. 4). (d) QPT is used to characterize the performance of the arbitrary holonomic gates. Arbitrary initial states are prepared with sequential pulses on $|g⟩\leftrightarrow |e⟩$ and $|e⟩\leftrightarrow |f⟩$ transitions. Nine sequential prerotation pulses on the qubit are performed before the final measurement to obtain the transmon’s full state tomography with three levels.

Figure 2

QPT of the single-qubit holonomic gates. (a) Unattenuated $\chi$ matrix fidelity $F_{\text{unatt}}$ of the single-qubit holonomic gates $U_1(\theta ,\gamma ,\varphi )$ with different $\theta$ and $\gamma$ while $\varphi =0$. The averaged fidelity ${\overline{F}}_{\text{unatt}}=0.994$ while the averaged attenuated fidelity ${\overline{F}}_{\mathrm{att}}=0.975$. (b) The traces of the reduced process matrix ${\chi }_r$ as a function of both $\theta$ and $\gamma$ with $\varphi =0$. The averaged trace is 0.992, indicating small leakage outside the computation subspace $\{|g⟩,|f⟩\}$. (c) Bar charts of the real and imaginary parts of ${\chi }_r$ of four specific gates: $X_{\pi }=U_1(\pi /2,\pi ,0)$, $X_{\pi /2}=U_1(\pi /2,\pi /2,0)$, $H=U_1(\pi /4,\pi ,0)$, and $Z_{\pi }=U_1(0,\pi ,0)$, where $R_{\phi }$ denotes a rotation of the qubit by an angle $\phi$ along the axis $R$ and $H$ represents the Hadamard gate. The numbers in the $x$ and $y$ axes correspond to the operators in the basis set $\{I,X,-iY,Z\}$ of the $\{|g⟩,|f⟩\}$ subspace. The solid black outlines are for the ideal gates.

Figure 3

RB of the single-qubit holonomic gates. (a) Sequences of both a reference RB experiment and an interleaved RB experiment. (b) The sequence fidelity decay as a function of the gate length $m$. The fidelity for each sequence length $m$ is measured for $k=100$ different random sequences with the standard deviation from the mean plotted as the error bars. Both curves are fitted with $F=A{p}^m+B$ with different sequence decays $p=p_{\mathrm{ref}}$ and $p_{\mathrm{gate}}$. The average gate fidelity per Clifford gate is $F_{\mathrm{avg}}=1-(1-p_{\mathrm{ref}})/2=0.996$. The difference between the reference and the interleaved RB experiments gives the specific gate fidelity $F_{\mathrm{gate}}=1-(1-p_{\mathrm{gate}}/p_{\mathrm{ref}})/2$. The red dashed line indicates the threshold for exceeding gate fidelity of 0.990.

Figure 4

Holonomic gates for cavity Fock state subspace $\{|0⟩,|1⟩\}$. (a) Illustration of the holonomic gates for Fock states, as well as the encoding and decoding processes. The holonomic gates are implemented by using a selective two-photon transition drive ${\mathrm{\Omega }}_1{e}^{i{\omega }_{d}t}$ on qubit $Q_1$ and a cavity-assisted Raman transition drive ${\mathrm{\Omega }}_2{e}^{i{\omega }_{p}t}$ between $|1g⟩$ and $|0f⟩$. The encoding and decoding processes are realized with a swap operation between qubit $Q_2$ and the cavity mode through a Raman transition drive similar to that for the gate. (b) Experimental sequence to perform QPT of the holonomic gates in the Fock $\{|0⟩,|1⟩\}$ subspace. (c) Bar charts of the real and imaginary parts of the process $\chi$ matrices for the whole process, including both the encoding and decoding processes, with the following gates: $X_{\pi }=U_2(\pi /2,0)$, $Y_{\pi }=U_2(\pi /2,\pi /2)$, $H_1=U_2(\pi /4,0)$, and $H_2=U_2(\pi /4,\pi /2)$. The attenuated and unattenuated process fidelities for these gates are 0.880, 0.881, 0.877, 0.879 and 0.990, 0.990, 0.970, 0.962, respectively. The numbers in the $x$ and $y$ axes correspond to the operators in the basis set $\{I,X,-iY,Z\}$. The solid black outlines are for the ideal gates.