High-fidelity all-microwave CZ gate with partial erasure-error detection via a transmon coupler

TL;DR

This work tackles the challenge of implementing high-fidelity two-qubit gates for quantum error correction in superconducting qubits using all-microwave control. It introduces the Transmon-Induced Phase (TIP) gate, which uses a fixed-frequency transmon coupler and multi-path coupling to suppress residual ZZ interactions while enabling fast CZ operations outside the straddling regime; the gate operates via a state-dependent dispersive shift in a gf-eg transition and requires a simple two-path geometric-phase evolution. The authors demonstrate a CZ gate with 99.7(1)% fidelity in 140 ns and show that approximately 45(4)% of two-qubit gate errors are detectable as erasures through mid-circuit coupler readout (PED), aligning with analytic error models. This combination of high-fidelity all-microwave control and hardware-level erasure detection provides a practical path toward erasure-aware quantum error correction in scalable superconducting processors.

Abstract

Entangling gates between neighboring physical qubits are essential for quantum error correction. Implementing them in an all-microwave manner simplifies signal routing and control apparatus of superconducting quantum processors. We propose and experimentally demonstrate an all-microwave controlled-Z (CZ) gate that achieves high fidelity while suppressing residual ZZ interactions. Our approach utilizes a fixed-frequency transmon coupler and multi-path coupling, thereby sufficiently reducing the net transverse interaction between data transmons to suppress residual ZZ interactions. The controlled phase arises from the dispersive frequency shift of the $|gf\rangle$$\unicode{x2013}$$|eg\rangle$ transition between the coupler and one of the data transmons conditioned on the state of the other data transmon. Driving the transitions at the midpoint of two dispersively shifted resonance frequencies induces state-dependent geometric phases to achieve the CZ gate. Crucially, with this scheme, we can maintain a small net transverse interaction between two data transmons while increasing the coupling between the data and coupler transmons to accelerate the CZ-gate speed. Additionally, we measure the coupler state after the gate to detect a subset of decoherence-induced failures that occur during the gate operation. These events constitute erasure errors with known locations, enabling erasure-aware quantum error-correcting codes to improve future logical qubit performance.

Figures (18)

• Figure 1: Transmon-induced phase (TIP) gate. (a) Schematic of the system. Two data transmons Q$_\mathrm{a}$ and Q$_\mathrm{b}$ interact transversely with the coupler transmon Q$_\mathrm{c}$ with the strengths $g_\mathrm{ac}$ and $g_\mathrm{bc}$, respectively. The direct interaction strength between the data qubits is $g_\mathrm{ab}$. (b) Energy-level diagram of the three-transmon system. The computational subspace is shaded yellow, and levels involving excitation of the coupler transmon are shaded green. Red and blue arrows indicate the $\ket{gf} \text{--} \ket{eg}$ transitions between Q$_\mathrm{b}$ and Q$_\mathrm{c}$ conditioned on the state of Q$_\mathrm{a}$. Green dashed lines indicate the Q$_\mathrm{a}$-dependent frequency shift of the $\ket{gf} \text{--} \ket{eg}$ transition, $\Delta_{gfeg}$. (c) Bloch-sphere depiction of the ideal evolution of the $\ket{gf} \text{--} \ket{eg}$ transitions in subspaces labeled by the state of Q$_\mathrm{a}$. The red trajectory corresponds to the case where $\mathrm{Q_a}$ is in $\ket{g}$, and the blue trajectory is for $\mathrm{Q_a}$ in $\ket{e}$. (d) Schematic spectra of the $\ket{gf} \text{--} \ket{eg}$ transition. When Q$_\mathrm{a}$ is in $\ket{e}$, the $\ket{gf} \text{--} \ket{eg}$ transitions frequency shifts downward in the case of $\omega_{ggh}>\omega_{egf}$. In the ideal case, the drive frequency $\omega_\mathrm{d}$ is set to the midpoint between the two resonance peaks.
• Figure 2: Device structure. (a) Optical micrograph of the fabricated three-transmon device (inset: Al/AlO$_x$/Al Josephson junction). Most features (yellow) use domain-matched epitaxially grown titanium-nitride (TiN) electrodes DMTiN on a Si substrate (gray). (b) Equivalent circuit of the coupled-transmon system. Q$_\mathrm{a}$, Q$_\mathrm{b}$, and Q$_\mathrm{c}$ denote two data transmons and a coupler transmon, respectively. Each transmon has its own drive line. R$_\mathrm{a}$, R$_\mathrm{b}$, and R$_\mathrm{c}$ denote the readout resonators, which connect with a common Purcell filter for multiplexed readout.
• Figure 3: Characterization of residual interaction. (a) Joint amplification of ZZ (JAZZ) pulse sequence that measures the ZZ interaction between the data transmons JAZZ0JAZZ1. The sequence is repeated with the roles of the measured data transmons exchanged. To improve the fitting accuracy, the phase of the second half-$\pi$ pulse ($\Phi_{\pi/2}$) is swept with the idling time $\tau$ such that $\phi/\tau = \omega_\mathrm{m}$. (b) Oscillation frequencies extracted from the JAZZ sequence. The frequencies are obtained by fitting the experimental data to a decaying cosine. The upper and lower panels show the results for $\mathrm{Q_a}$ and $\mathrm{Q_b}$, respectively. Note that the vertical axis for Q$_\mathrm{b}$ is inverted in the bottom half. Dashed lines indicate linear fits to the positive- and negative-slope branches. The residual ZZ interaction, averaged over $\mathrm{Q_a}$ and $\mathrm{Q_b}$, is indicated by the black dashed line and by the red dot in the inset.
• Figure 4: Implementation of TIP gate. (a) Pulse sequence used to measure the $\ket{gf} \text{--} \ket{eg}$ Rabi oscillations between Q$_\mathrm{b}$ and Q$_\mathrm{c}$. To condition on the state of Q$_\mathrm{a}$, we use two variants of the Q$_\mathrm{a}$ sequence, with and without an X$_\pi$ pulse. A flat-top microwave pulse drives the $\ket{gf} \text{--} \ket{eg}$ transition, called a TIP pulse, with raised-cosine edges (edge length 20 ns). (b,c,d) $\ket{gf} \text{--} \ket{eg}$ Rabi-oscillation frequency $\Omega_{gfeg}^i$, resonance frequency $\omega_{gfeg}^i$, and the state-dependent frequency shift $\Delta_{gfeg}$. For each drive amplitude $A_\mathrm{d}$, we record Rabi oscillations by sweeping both the pulse duration $\tau$ and the drive frequency $\omega_\mathrm{d}$ of the TIP pulse and fit the patterns with an exponentially decaying cosine. From the fit, we extract the minimum $\ket{gf} \text{--} \ket{eg}$ Rabi-oscillation frequency for each subspace [plotted in (b)] and the corresponding drive frequency [plotted in (c)]. Panel (d) shows $\mathrm{Q_a}$-state-dependent frequency shift of $\ket{gf} \text{--} \ket{eg}$ resonance $\Delta_{gfeg} = \omega^e_{gfeg} - \omega^g_{gfeg}$. The dashed lines in (b) and (c) indicate linear and quadratic fits, respectively. In (d), the dashed line is the difference between the fits in (c). In (b), the horizontal black dotted line marks $\sqrt{3} \, \Delta_{gfeg}/2$ at zero drive amplitude, calculated from the fit. The vertical black dotted line marks the drive amplitude at which the fit for the case with $|g\rangle_\mathrm{a}$ intersects the horizontal line. The upper horizontal axes show the drive strength $\Omega_\mathrm{d}$, calibrated with the linear coefficient of the linear fit to the data for $|g\rangle_\mathrm{a}$ in (b). (e,f) $\ket{gf} \text{--} \ket{eg}$ Rabi-oscillation chevron patterns at the drive amplitude indicated by the vertical black dotted line in (b). Horizontal white dashed lines mark the condition that minimizes the $\ket{f}_\mathrm{c}$-state population simultaneously across the subspaces corresponding to $\ket{g}_\mathrm{a}$ and $\ket{e}_\mathrm{a}$ after one full cycle of the Rabi oscillation, determined from exponentially decaying-cosine fits. (g) Rabi oscillations for Q$_\mathrm{b}$ and Q$_\mathrm{c}$ in each subspace under the drive-frequency condition indicated by the white dashed lines in (e) and (f). The dashed curves show fits to an exponentially decaying cosine. The vertical black dotted line denotes the initial estimate of the flat-top duration $\tau$ of the TIP pulse implementing the CZ gate.
• Figure 5: Optimization of TIP gate. (a) JAZZ2-N pulse sequence. Here, we use $N = 2k$, and $k=5$. (b) Evolution of the state populations $P(|ggg\rangle)$, $P(|e\rangle_\mathrm{c})$, and $P(|f\rangle_\mathrm{c})$, corresponding respectively to the states $\ket{ggg}$, $\ket{e}_\mathrm{c}$, and $\ket{f}_\mathrm{c}$, during the optimization process using the JAZZ2-N pulse sequence. The optimization cost is $P(|ggg\rangle)$. As the optimization proceeds, $P(|f\rangle_\mathrm{c})$ decreases, indicating suppression of the leakage into $\ket{f}_\mathrm{c}$ state, while $P(|e\rangle_\mathrm{c})$ remains unchanged, confirming that population leakage into $\ket{e}_\mathrm{c}$ state is not increasing. The optimization is performed using the CMA-ES algorithm implemented in Optuna optuna, with a population size of 15 per generation. After 50 generations, the parameter set used for subsequent experiments is taken as the mean of the parameters from the final generation. (c,d,e) Convergence behavior of the drive amplitude $A_\mathrm{d}$, drive frequency $\omega_\mathrm{d}$, and flat-top pulse duration $\tau$, during the optimization process. The flat-top duration takes discrete values due to the time-resolution granularity of the AWGs used in the experiment. (f) Pulse sequence to calibrate the local phases induced by the TIP pulse on each data transmon. In the calibration protocol, the number of TIP pulses, $L$, is increased, and tomography is performed along the $x$- and $y$-axes to measure the accumulated phase. (g) Measured accumulated phases induced by the TIP pulse on each data transmon. The dashed lines represent linear fits, and the slope extracted from each fit determines the virtual-Z-gate phase required for the CZ-gate implementation.