Leakage reduction in fast superconducting qubit gates via optimal control

Superconducting transmon qubits are promising for scalable quantum computing due to mature microwave control and stable operations. Achieving high-speed, high-fidelity gates is challenging because weak anharmonicity causes leakage from the computational subspace under short, broadband pulses. Open-loop optimal control can, in principle, deliver high-fidelity state preparation and gates given accurate models, but in superconducting platforms model inaccuracies (e.g., instrument noise, transfer functions, extra modes, unintended couplings) limit performance relative to other systems. Consequently, closed-loop optimization on the experimental device is often necessary, which constrains the number of tunable pulse parameters. Reducing leakage is critical for error correction because leakage errors are far more resource-intensive to correct than errors within the computational subspace. This work addresses the problem of designing fast single-qubit gates with minimal leakage by implementing a closed-loop optimal control protocol that directly optimizes digitized pulse samples to overcome model limitations and enhance fidelity at very short durations.

The paper builds on prior developments in optimal control for quantum systems, including open-loop strategies that have been successful in ion traps and NMR but are less effective for superconducting qubits due to imperfect models. DRAG pulses mitigate leakage and phase errors from the 1→2 transition but degrade below approximately 10/Δ in duration. Prior works have addressed frequency crowding, cross-talk mitigation, and readout/reset speedups. Randomized benchmarking has been used as a robust calibration and characterization tool. The authors leverage and extend these ideas by using closed-loop optimization with a large parameter set, enabled by efficient measurement protocols and a population-based optimizer suited to noisy evaluations.

• System and control: Experiments were performed on one transmon of a two-qubit fixed-frequency device coupled via a tunable coupler. The qubit frequency is 5117.22 MHz with anharmonicity −315.28 MHz and coherence times T1 = 105 µs and T2 = 39 µs. Microwave control is applied via the readout resonator using single-sideband upconversion at 100 MHz. Pulses are generated by a 2.4 GS/s AWG, providing sample-wise control of the in-phase and quadrature components.
• Pulse parameterization: Starting from an analytically calibrated DRAG Gaussian pulse with first-order derivative correction, the authors introduce piecewise-constant sample-wise corrections to both quadratures. For a pulse of N samples, per-sample real and imaginary corrections (a_n, b_n) are added to the DRAG samples, yielding a pulse fully defined in AWG memory.
• Closed-loop optimization: All parameters are optimized simultaneously using the CMA-ES algorithm, which evaluates populations of candidate parameter sets and iteratively updates the sampling distribution. A cost function based on randomized benchmarking (RB) is used: for each candidate pulse shape, Clifford sequences of fixed length m are constructed from ±X/2 and ±Y/2 pulses defined by that shape. The average ground-state population p0(m) over K random sequences serves as the cost to maximize. This single-point RB approach reduces runtime compared to full RB curves. Restless measurement enables a 100 kHz repetition rate. The optimization proceeds in two stages: (1) calibrate DRAG parameters A, β, and ω_ssb by CMA-ES; (2) initialize from the optimized DRAG and extend the parameter set to include all sample-wise corrections {a_n, b_n} for full piecewise-constant (PWC) optimization.
• Experimental settings: Pulses with N = 10–26 samples (durations 4.16–10.83 ns) were optimized. For cost evaluation they used K = 20 sequences with m = 120 Cliffords, each sequence measured 1000 times using restless readout. Drive power limited minimum gate duration to ~4 ns.
• Leakage characterization: For the shortest pulse (4.16 ns), leakage randomized benchmarking was performed by measuring populations p_j in states |0>, |1>, |2> after RB sequences. The computational-subspace population px = p0 + p1 − p2 was fit to A + B λ1^n to extract leakage behavior. Using the leakage decay term, p0(n) was fit to a double decay A0 + B λ1^n + C λ2^n to obtain the average Clifford fidelity F = 0.5 [λ1^2 + 1 − λ1].
• Runtime analysis: They profiled per-iteration time into (i) pulse sequence processing/compilation to AWG files, (ii) hardware initialization and data transfer, and (iii) data acquisition. Due to constant initialization overheads, population-based optimizers like CMA-ES are more efficient than single-point optimizers (e.g., Nelder–Mead). Restless measurements keep acquisition time small. Optimizing the longest pulse (N = 26, 55 parameters) took up to 25 hours; improvements such as internal sideband generation could reduce runtime.
• Numerical simulations: For comparison, the qubit is modeled as a driven anharmonic oscillator (d = 4) with T1 and T2 included via Lindblad master equations. Superoperators for ±X/2 and ±Y/2 are composed to form Clifford operations. L-BFGS gradient optimization is used numerically (noise-free) to maximize average Clifford fidelity. Simulations help identify T1 limits and effects of amplitude/phase noise in control electronics.

• Achieved a 4.16 ns single-qubit pulse with average fidelity per Clifford of 99.76(8)% and leakage rate 0.044(25)%. This corresponds to a sevenfold reduction in leakage and a threefold reduction in standard errors compared to the best calibrated DRAG pulse at similar duration.
• For gate durations >5 ns, both DRAG and PWC pulses achieve constant fidelity of 99.87(1)% (error per gate ~0.13%). For durations <5 ns, DRAG fidelities degrade consistent with the ~10/Δ limit, while PWC-optimized pulses maintain high fidelity down to 4.16 ns.
• Leakage RB for the 4.16 ns pulses: LPWC = 0.044(25)% versus LDRAG = 0.29(3)%. Standard errors reduced from 1 − ADRAG = 1.49(15)% to 1 − APWC = 0.44(15)%. Using the double-decay model yields FPWC = 99.76(8)% and FDRAG = 99.11(8)%.
• Experimental fidelities are not T1-limited (T1 error per gate limit ~5×10^-5). The observed noise floor is consistent with control amplitude and/or phase noise; PWC optimization mitigates unknown error sources present at high drive power where DRAG underperforms.
• Calibration feasibility: Closed-loop optimization handling up to 55 parameters was demonstrated, though with substantial runtime (up to 25 h for N = 26).

Optimizing pulses in a piecewise-constant basis via closed-loop CMA-ES effectively suppresses leakage and maintains high fidelities for very short single-qubit gates, surpassing analytical DRAG performance in the fast-gate regime. This is particularly impactful where single-qubit gate times approach two-qubit gate durations, helping reduce overall algorithm wall time and error accumulation. While additional pulse complexity does not improve long-duration gates beyond ~0.13% error per gate, the method crucially overcomes model inaccuracies and high-power nonidealities that limit open-loop designs. The achieved fidelities are not limited by T1, indicating residual control noise as the dominant limitation; simulations support that amplitude/phase noise in control electronics can explain the fidelity floor. The work also highlights practical considerations: population-based optimizers are better matched to experimental overheads, and instrument improvements (e.g., internal sideband generation) could significantly reduce calibration time. The demonstrated feasibility of optimizing tens of parameters opens avenues for extending closed-loop optimal control to more complex multi-qubit operations, where system dynamics are richer and analytical solutions less mature.

The study demonstrates that closed-loop optimal control with sample-wise pulse optimization substantially reduces leakage and improves fidelity for fast single-qubit gates in superconducting transmons. A 4.16 ns gate achieved 99.76% fidelity with 0.044% leakage, outperforming DRAG at short durations by reducing leakage sevenfold and standard errors threefold. For gate durations above ~5 ns, both DRAG and optimized PWC pulses reach similar high fidelities (~99.87%). The approach is experimentally viable with up to 55 parameters, though calibration time is significant. Future work will target multi-qubit gates where more expressive pulse parameterizations (e.g., spectral representations) may be advantageous, and hardware improvements are expected to reduce optimization runtime and further enhance performance.

• Experimental fidelities are limited by control noise (amplitude/phase) rather than T1, with exact noise composition not fully identified.
• Drive power constraints prevented implementing gates shorter than ~4 ns.
• Calibration is time-consuming (up to 25 hours for 55 parameters), with significant overhead from sequence processing and hardware initialization.
• Open-loop models lack sufficient accuracy for immediate high performance on superconducting qubits, necessitating closed-loop optimization.
• Extending piecewise-constant parameterization to longer two-qubit gates is more challenging due to increased duration and complexity.