Universal Quantum Gate Set Approaching Fault-Tolerant Thresholds with Superconducting Qubits

We use quantum process tomography to characterize a full universal set of all-microwave gates on two superconducting single-frequency single-junction transmon qubits. All extracted gate fidelities, including those for Clifford group generators, single-qubit $\pi /4$ and $\pi /8$ rotations, and a two-qubit controlled-not, exceed $95\%$ ($98\%$), without (with) subtracting state preparation and measurement errors. Furthermore, we introduce a process map representation in the Pauli basis which is visually efficient and informative. This high-fidelity gate set serves as a critical building block towards scalable architectures of superconducting qubits for error correction schemes and pushes up on the known limits of quantum gate characterization.

Figure 1

Single-junction transmon and quantum process tomography pulse sequence. (a) Optical micrograph of a single-junction transmon device shown coupled to the center pin of a microwave resonator. The design is similar to standard transmon devices, with interdigitated capacitors shunting the $\mathrm{Al}/{\mathrm{AlO}}_x$ Josephson junction on either side. Inset: scanning-electron micrograph of the Josephson junction with critical current $I_0=28\mathrm{nA}$. (b) Pulse sequence for performing quantum process tomography. An overcomplete set of rotations $\{I,X_{\pi },X_{\pm \pi /2},Y_{\pm \pi /2}\}$ is used to generate input states and to analyze the process before joint readout via the cavity.

Figure 2

Quantum process tomography for $X_{\pi /4}\otimes I$ represented as the Pauli transfer matrix $\mathcal{R}$. (a) Experimentally extracted ${\mathcal{R}}_{\mathrm{mle}}$ for $X_{\pi /4}\otimes I$ with gate fidelity $F_g=0.968(7)$. To illustrate the action of ${\mathcal{R}}_{\mathrm{mle}}$, an input state $|00⟩$ in the Pauli state vector form ${\overrightarrow{p}}_{\mathrm{in}}$ is shown above ${\mathcal{R}}_{\mathrm{mle}}$, and the output state $\cos (\pi /8)|00⟩-i\sin (\pi /8)|10⟩$ ($F_s=0.997$), represented as ${\overrightarrow{p}}_{\mathrm{out}}$, is shown to the right. (b) Ideal ${\mathcal{R}}_{\mathrm{ideal}}$ for $X_{\pi /4}\otimes I$.

Figure 3

Quantum process tomography for the ${\mathrm{CNOT}}_{12}$ operation represented as the Pauli transfer matrix $\mathcal{R}$. (a) Experimentally extracted ${\mathcal{R}}_{\mathrm{mle}}$ for the ${\mathrm{CNOT}}_{12}$ operation with gate fidelity $F_g=0.9507$. To illustrate the action of ${\mathcal{R}}_{\mathrm{mle}}$, an input state $(|0⟩+i|1⟩)(|0⟩+i|1⟩)/2$ represented as ${\overrightarrow{p}}_{\mathrm{in}}$ is shown above ${\mathcal{R}}_{\mathrm{mle}}$, and the output entangled state $(|00⟩+i|01⟩-|10⟩+i|11⟩)/2$ ($F_s=0.9827$, $\mathcal{C}=0.994$), represented as ${\overrightarrow{p}}_{\mathrm{out}}$, is shown to the right. (b) Ideal ${\mathcal{R}}_{\mathrm{ideal}}$ for the ${\mathrm{CNOT}}_{12}$ operation.

Figure 4

Concurrence and state fidelity after $N$ applications of the CNOT operation. (a) Concurrence $\mathcal{C}$ of final states after applying $N$ CNOT gates to input state $(|00⟩+i|10⟩)/\sqrt{2}$. (b) Black squares are the final state fidelity $F_s$ of experimentally obtained states to the ideal Bell (original input) state for odd (even) $N$. The dashed black line is a fit to the data assuming a model of $A{F}_g^N+B$, where $A$, $B$, and gate fidelity $F_g$ are fit parameters. The fit gives an error per gate $1-F_g=0.0164$. The solid blue line is a theoretical model using the measured coherence values and gate times.