Realizing a Continuous Set of Two-Qubit Gates Parameterized by an Idle Time

TL;DR

The paper addresses the challenge of implementing hardware-efficient, continuously parameterizable two-qubit gates that remain robust to pulse distortions in near-term devices. It extends net-zero $C Z_{\pi}$ gates to a continuous $C Z_{\theta}$ family by using two half-waveforms separated by an idle time $t_{\mathrm idle}$, yielding a linear relation $\theta = \pi - \theta_{\mathrm idle}$ with $\theta_{\mathrm idle} = \int \Delta(t)\,dt$, and it relies on resonance tuning between $|11\rangle$ and $|20\rangle$ to minimize leakage across the whole gate set. The authors introduce a leakage-amplification method to coherently measure leakage and a modified cross-entropy benchmarking (XEB) cycle to benchmark weakly entangling gates, validating uniform fidelity across $\theta$ with an average gate error around $0.61\%$ and per-gate leakage in the $2\times 10^{-4}$ to $4\times 10^{-4}$ range on a fixed-coupling six-transmon device. The result is a practical, one-parameter gate-set calibration that can reduce circuit depth for near-term quantum algorithms and potentially improve the feasibility of quantum error correction by enabling low-leakage, high-fidelity two-qubit operations.

Abstract

Continuous gate sets are a key ingredient for near-term quantum algorithms. Here, we demonstrate a hardware-efficient, continuous set of controlled arbitrary-phase ($\mathrm{C}Z_θ$) gates acting on flux-tunable transmon qubits. This implementation is robust to control pulse distortions on time scales longer than the duration of the gate, making it suitable for deep quantum circuits. Our calibration procedure makes it possible to parameterize the continuous gate set with a single control parameter, the idle time between the two rectangular halves of the net-zero control pulse. For calibration and characterization, we develop a leakage measurement based on coherent amplification, and a new cycle design for cross-entropy benchmarking. We demonstrate gate errors of $0.7 \%$ and leakage of $4\times 10^{-4}$ across the entire gate set. This native gate set has the potential to reduce the depth and improve the performance of near-term quantum algorithms compared to decompositions into $\mathrm{C}Z_π$ gates and single-qubit gates. Moreover, we expect the calibration and benchmarking methods to find further possible applications.

Figures (16)

• Figure 1: Concept of the gate set. a Control pulses, as a function of time (vertical axis), in units of magnetic flux at the SQUID loop of each transmon. b Qubit transition frequencies $\omega_{10}$ and $\omega_{01}$ of the two transmons as a function of flux. c,d Frequencies of the relevant two-transmon states (d) and detuning $\Delta(t) = \omega_{20}-\omega_{11}$ (c) over time due to the flux pulses in a and the conversion to frequencies in b. The shaded area in c indicates the phase $\theta_{\mathrm{idle}}$ acquired by the $\ket{20}$ state during the idle time $t_{\mathrm{idle}}$. e,f Measured population (brightness) and conditional phase (hue) of the $\ket{11}$ state at the end of the gate, see measurement techniques in the main text. Shown are two cuts through the 3-dimensional parameter space ($\Delta_{\mathrm{min}}$, $\theta_{\mathrm{idle}}$, $t_{\mathrm{int}}$), at $\theta_{\mathrm{idle}}=0$ (e) and $\Delta_{\mathrm{min}}=0$ (f). The white point indicates a $\mathrm{C}Z_{\pi}$ gate, the dashed line the continuous gate set. g Bloch sphere illustrating state trajectories for several $\Delta_{\mathrm{min}}$ yielding $\theta\in\{\pi/4, \pi/2, ..., 7\pi/4\}$, for $\theta_{\mathrm{idle}}=0$ as in e (idealized illustration assuming pulses with zero rise time). h Illustration of the implementation of the continuous gate set by means of a variable idle phase as in f, see main text.
• Figure 2: Characterization of the resonance condition. a Population recovery and conditional phase (color scheme as in \ref{['fig:1']}) as a function of the detuning $\Delta_{\mathrm{min}}$ (controlled by the pulse amplitude $\phi_{\mathrm{low}}$) and interaction time $t_{\mathrm{int}}$. For these measurements, $\theta_{\mathrm{idle}}$ is set to $\pi$, resulting in maximum population sensitivity to $\Delta_{\mathrm{min}}$ (illustrated by the white arrow). b Pulse sequence for coherent amplification of leakage, by repeating $N$ times the flux pulses (FP) implementing the gate. Repetition is denoted by vertical bars with dots, and $t_{\mathrm{sep}}$ is the time between centers of the falling and rising edges of successive gates. c Second excited state population, $P_{\mathrm{Q_{high}},2}$, of $\mathrm{Q_{high}}$, color-coded according to the color scale in panel d, measured after $N=16$ gates as a function of flux amplitude $\phi_{\mathrm{low}}$ and gate separation $t_{\mathrm{sep}}$ (purple and black arrows in b). A line cut for a fixed $\phi_{\mathrm{low}}$ is included in \ref{['supp:la']}. d Maximum of $P_{\mathrm{Q_{high}},2}$ (scale on the left vertical axis), taken over all gate separations $t_{\mathrm{sep}}$, at each flux amplitude $\phi_{\mathrm{low}}$. The approximate conversion to the leakage $L$ of a single gate, shown on the right axis, is given in \ref{['supp:la']} (valid outside the red shaded area).
• Figure 3: Gate set calibration and benchmarking. a Conditional phase calibration as a function of $t_{\mathrm{idle}}$ (colored as in \ref{['fig:1']}), measured using the method detailed in \ref{['supp:calib']}, and linear fit (black line) with residuals (gray). b XEB quantum circuit, consisting of $M$ successive cycles. Each cycle $i$ comprises a $\mathrm{C}Z_{\theta_i}$ gate with phase $\theta_i$, and a mixing unitary composed of randomly chosen single-qubit gates (squares) and a $\mathrm{C}Z_{\pi}$ gate. c Gate error from XEB measurements, for 1750 random circuits of up to $M=256$ cycles, for a fixed phase $\theta_i=\theta$ across all cycles (colored as in \ref{['fig:1']}), and for a random phase $\theta_i$ in each cycle (dashed gray line). The result of interleaved randomized benchmarking (IRB) is shown in black. d Leakage extracted from XEB and IRB measurements (legend as in c).
• Figure 4: False-color micrograph of the device used in a preliminary version of this experiment; the device used in the final experiment is nominally identical. Each transmon qubit consists of a capacitive island (yellow) connected to the ground plane via a superconducting quantum interference device (SQUID), and capacitively coupled to a drive line (pink) and the readout circuitry (purple). Fast flux control is provided by individual flux lines (green), which are inductively coupled to the SQUID of each qubit. The capacitive qubit-qubit coupling is implemented with fixed-frequency coupling resonators (light blue). The measurements have been performed on the pair of qubits indicated by a white frame.
• Figure 5: Simplified schematic of the experimental setup, including the quantum device. Coupling elements between transmons are not represented. See the text for details.