High-Precision Calibration Workflow Achieves Above $99.9\%$ CZ Gate Fidelity on a Scalable Superconducting Processor

Abstract: High-fidelity universal two-qubit gates are critical for building fault-tolerant quantum computers. In scalable superconducting processors, shortened coherence times introduce more incoherent errors in gate operations. With a constrained error budget, there is reduced tolerance for coherent errors stemming from parameter deviations. In this work, we develop a closed-loop workflow to enhance the CZ gate calibration precision. Utilizing the echoed leakage error amplification (ELEA) and the repurposed context-aware fidelity estimation (CAFE) circuits, we suppress the population leakage to non-computational states, and, for the first time, demonstrate a CZ gate fidelity exceeding $99.9\%$ on an 84-qubit processor, with coherent error suppressed to $0.007\%$. Meanwhile, we obtain a median fidelity of $99.25\%$ among 72 CZ gates, demonstrating that the workflow can be generalized to the calibration of parallel CZ gates. Finally, we realize automated calibration and observe enhanced stability of the CZ gate throughout 9-hour comparative monitoring experiments. Our results, realized on a completely domestic platform, establish an efficient and automated route to quantum computation with superconducting quantum systems.

• The paper presents a closed-loop workflow combining SEA, ELEA, and CAFE calibration to achieve 99.92 ± 0.01% CZ fidelity on the best qubit pair while suppressing coherent errors to 0.007%.
• Parallel calibration across 72 CZ gates achieved a 99.25% median fidelity, showing that relaxation and dephasing—not calibration precision—dominate processor-wide errors.
• Automated recalibration every 30 minutes sustained lower CZ error rates over nine hours, demonstrating a practical path to reliable operation despite environmental drift.

Overview

This paper reports a closed-loop calibration workflow for controlled-Z (CZ) gates on the 84-qubit superconducting processor Shenglian, composed of tunable transmon qubits coupled via tunable transmon couplers (2607.01422). The central result is a CZ gate fidelity of $(99.92\pm0.01)\%$ on a scalable processor, with coherent errors suppressed to $0.007\%$, alongside a median fidelity of $(99.25\pm0.01)\%$ across 72 parallel-calibrated CZ gates. The authors emphasize that prior demonstrations of sub-0.1%-error CZ gates were achieved on isolated samples, whereas scaling to large processors introduces additional incoherent error channels—two-level system (TLS) defects at enlarged metal-substrate interfaces and depolarizing channels from denser control wiring—that shrink the error budget available for coherent errors. The workflow addresses precisely this constraint.

Calibration workflow

The workflow operates on a minimal three-element unit: two nearest-neighbor qubits $Q_a$ ($\omega_a/2\pi \approx 4.224$ GHz) and $Q_b$ ($\omega_b/2\pi \approx 4.411$ GHz) coupled through tunable coupler $C$. The coupler is biased at $\omega_C/2\pi \approx 3.2$ GHz, where both effective $XY$ and residual $0.007\%$0 interactions are minimized; next-nearest-neighbor interactions below $0.007\%$1 MHz are neglected. Single-qubit gates use 14 ns raised-cosine pulses with DRAG correction, yielding randomized benchmarking (RB) errors per Clifford of $0.007\%$2 and $0.007\%$3 for $0.007\%$4 and $0.007\%$5, respectively.

The CZ gate is implemented via the non-adiabatic scheme, tuning both qubits into resonance with $0.007\%$6 while the coupler enhances the $0.007\%$7–$0.007\%$8 coupling; a full oscillation yields the conditional $0.007\%$9 phase. Flux pulses follow a hyperbolic-cosine shape with steepness $(99.25\pm0.01)\%$0, duration $(99.25\pm0.01)\%$1 ns, plus a 16 ns buffer on each side. Three pulse amplitudes—$(99.25\pm0.01)\%$2, $(99.25\pm0.01)\%$3 for the qubits and $(99.25\pm0.01)\%$4 for the coupler—are calibrated in a staged procedure:

1. Coarse stage: a two-dimensional scan of $(99.25\pm0.01)\%$5, $(99.25\pm0.01)\%$6 at fixed modest $(99.25\pm0.01)\%$7 identifies regions of accumulated conditional phase on $(99.25\pm0.01)\%$8.
2. Intermediate stage: standard error amplification (SEA) and echoed leakage error amplification (ELEA) circuits scan $(99.25\pm0.01)\%$9 and $Q_a$0 alternately, with convergence as the amplified gate number increases.
3. Fine stage: three-state single-shot readout is characterized, and the repurposed context-aware fidelity estimation (CAFE) circuit replaces direct $Q_a$1 calibration by measuring $Q_a$2 population over 60 repeated CZ gates.
4. Evaluation: if fidelity falls below a threshold $Q_a$3—relevant given frequency crowding and collisions with neighboring qubits or TLS defects—the choice of $Q_a$4, $Q_a$5 is updated and calibration repeats.

The ELEA circuit is a modification of the phase-averaged leakage error amplification (PALEA) circuit of Marxer et al.: by inserting an additional CZ gate and two $Q_a$6 gates per repetition unit, it eliminates the need to randomize the phase $Q_a$7 of the $Q_a$8 leakage-refocusing pulse, simplifying implementation without loss of leakage sensitivity.

Benchmarking results

Standard and interleaved RB on the best pair yield an error per Clifford of $Q_a$9 (averaging 1.5 CZ gates and 5 single-qubit gates per Clifford), corresponding to an interleaved CZ fidelity of $\omega_a/2\pi \approx 4.224$0. A duration sweep reveals an explicit trade-off: durations from 30 to 48 ns all achieve errors below $\omega_a/2\pi \approx 4.224$1, but shorter gates degrade because the coupler must be tuned closer to the qubits, inducing leakage to the coupler, while residual short-time flux-pulse distortion introduces coherent errors. This trade-off bounds how much further incoherent error can be reduced by speed alone.

CAFE-based error decomposition attributes $\omega_a/2\pi \approx 4.224$2 to decoherent error and only $\omega_a/2\pi \approx 4.224$3 to coherent error, giving an estimated fidelity of $\omega_a/2\pi \approx 4.224$4 consistent with IRB. Leakage to $\omega_a/2\pi \approx 4.224$5 is bounded at $\omega_a/2\pi \approx 4.224$6. A dynamical-decoupling variant (DCAFE), interleaving $\omega_a/2\pi \approx 4.224$7 gates after each CZ pair, echoes out low-frequency noise and reduces the coherent component further to $\omega_a/2\pi \approx 4.224$8—confirming that the residual coherent error is dominated by correctable single-qubit phase errors rather than intrinsic two-qubit imperfections.

Across the processor, 65 single-qubit and 72 CZ gates were calibrated in parallel. Median errors are $\omega_a/2\pi \approx 4.224$9 (single-qubit) and $Q_b$0 (CZ). The paper establishes a clear physical correlation: single-qubit error rates scale exponentially with inverse dephasing time, $Q_b$1, and CZ error rates correlate spatially with single-qubit errors, both tracing to short dephasing times. For the full ensemble, median error components decompose as $Q_b$2 coherent (SU(4) phase errors), $Q_b$3 leakage-induced, and $Q_b$4 from relaxation/dephasing—with the latter dominating by more than an order of magnitude. The implication is that further device-wide gains require improving coherence, not calibration precision; the calibration workflow has already reduced its controllable contributions to negligible levels.

Automated calibration

Because qubits and couplers operate away from sweet spots, gates are susceptible to environmental drift. In an 18-hour experiment on six qubit pairs, CZ error rates were tracked over 9 hours under automated recalibration every 0.5 hours versus a calibration-free scenario. Automated calibration lowers average error rates throughout, with the largest benefit on the most fluctuating pairs (e.g., $Q_b$5–$Q_b$6). This demonstrates that the fine-calibration step is lightweight enough to run continuously, supporting sustained high-fidelity operation on large processors.

Limitations and open questions

The headline $Q_b$7 fidelity is demonstrated on a single qubit pair; the fleet-wide median of $Q_b$8 remains roughly a factor of eight worse, limited predominantly by dephasing-limited coherence (median $Q_b$9 μs, $\omega_b/2\pi \approx 4.411$0 μs across 65 qubits). The paper does not demonstrate whether the best-pair performance can be reproduced broadly, nor does it quantify crosstalk during simultaneous operation of the six coupler groups beyond noting frequency-collision screening in the evaluation step. The duration trade-off between coupler-induced leakage and flux-pulse distortion leaves open which pulse-shape or spectral-engineering optimizations could push gate durations below 30 ns without penalty. Finally, the automated recalibration interval of 0.5 hours was chosen empirically; the optimal cadence relative to drift timescales is not established.

Conclusion

The paper demonstrates that careful, circuit-level error-amplified calibration—combining SEA, ELEA, and CAFE in a closed loop—can suppress coherent CZ gate errors to the $\omega_b/2\pi \approx 4.411$1 level on a scalable 84-qubit superconducting processor, achieving above-$\omega_b/2\pi \approx 4.411$2 fidelity where the remaining error budget is dominated by incoherent relaxation and dephasing. The workflow generalizes to parallel calibration, supports fully automated operation, and was executed entirely on domestically sourced instrumentation. Its principal contribution is methodological: a reproducible route to near-threshold two-qubit gate performance whose further improvement now hinges on materials and coherence engineering rather than calibration precision.