Optimal robust single-qubit gates in superconducting quantum processors

AI summary

Overview

Research area: Quantum physics — superconducting quantum computing, specifically pulse-level quantum control (optimal control and robust gate design) for single-qubit operations.

Technical level: Intermediate. The core ideas (robustness to control errors, pulse shaping, leakage suppression) are explained conceptually here, but the paper itself relies on perturbative inverse-engineering mathematics and randomized benchmarking, so a reader wanting the derivations needs a graduate-level control-theory background.

Scope (one sentence): The paper introduces and experimentally demonstrates "time-rescaling robust inverse-optimal" (TRIO) microwave pulses that implement single-qubit gates on a superconducting transmon with high fidelity and order-of-magnitude-improved tolerance to drive-amplitude and frequency-detuning errors.

What This Paper Is About

Fault-tolerant quantum computing needs single-qubit gates that stay accurate even when the hardware drifts: microwave pulses are miscalibrated, qubit frequencies shift, and weakly anharmonic qubits leak population out of the computational subspace. Existing "robust" pulse designs can suppress these errors, but the optimal waveforms they produce tend to have sharp, square-pulse-like edges that real microwave hardware cannot faithfully deliver, which in turn causes leakage.

The goal of this work is to keep the error-robustness and energy-optimality of those designs while producing smooth, hardware-compatible waveforms — and then to test them on an actual superconducting processor rather than only in simulation.

Key Contributions

1. A TRIO pulse protocol. The authors combine the robust inverse-optimal (RIO) framework with a time-rescaling technique. Because the Schrödinger evolution is invariant under a reparametrization of time, an optimized but abrupt Rabi envelope can be replaced by a smooth "hyper-Gaussian" envelope, with the detuning profile reconstructed so that the original energy-optimal geometric trajectory in the (θ, γ) plane is exactly reproduced.

2. Leakage suppression on top of robustness. The smooth TRIO waveform permits standard DRAG (derivative removal by adiabatic gate) corrections to be applied, retaining leading-order terms in 1/ξ (ξ being the transmon anharmonicity), which suppresses leakage to higher levels and compensates the AC Stark shift.

3. First experimental demonstration of optimal robust single-qubit gates. The pulses are implemented on a superconducting processor containing two fixed-frequency transmon qubits coupled by a flux-tunable coupler, using IQ control plus a compensating virtual-Z rotation whose phase is taken from the optimal-control solution and refined experimentally. Reconstructed detuning profiles are implemented via microwave phase modulation rather than flux tuning.

4. A universal single-qubit gate set with error statistics. Beyond the X gate, the framework yields a Hadamard-type operation (pulse area 3.395π, with φ = 0.73π) from which X_π/2 and Y_−π/2 are generated by adjusting virtual-Z rotations, and the full set is benchmarked under randomly sampled amplitude and detuning errors.

Main Findings

• High X-gate fidelity at 90 ns. Interleaved randomized benchmarking (IRB) gives X-gate fidelities of 99.90 ± 0.01% for the TRIO pulse and 99.94 ± 0.01% for the Gaussian pulse at t_f = 90 ns, averaged over independent experiments; the abstract states fidelities above 99.9%. The Gaussian pulse has slightly better peak fidelity.

• Robustness under combined errors. The TRIO X gate keeps fidelities above 99.5% for simultaneous drive-amplitude variations up to 20% and qubit detuning up to 4 MHz. Under the same combined errors the Gaussian pulse falls to roughly 92%. The Gaussian pulse tolerates about an order of magnitude smaller errors to maintain the same fidelity.

• Two-dimensional fidelity maps. The high-fidelity region for the Gaussian pulse is narrow and tilted, while the TRIO contours for F ≥ 99% and F ≥ 99.5% enclose a region an order of magnitude larger in both amplitude error α and detuning δ. Simulations including decoherence (measured T₁ = 75.4 µs, T₂* = 55.1 µs) agree well with the data.

• Simulated error levels. Numerical simulations for the X gate give gate errors of approximately 10⁻⁴ for TRIO versus 10⁻¹ for Gaussian pulses under α = 20% and δ = 4 MHz, with the TRIO trajectory deviating mid-evolution but reconverging near the target state.

• Process tomography. Quantum process tomography over ten independent experiments gives an average estimated X-gate fidelity of 99.93 ± 0.05%, with small deviations in the reconstructed Choi matrix relative to the ideal one.

• Hadamard-type gates. The TRIO-based H gate reaches an IRB fidelity of 99.63(1)%, compared with 99.92(1)% for a Gaussian-based implementation. The abstract reports Hadamard-type fidelities above 99.6%.

• Universal set benchmarks. IRB fidelities are 99.91(2)% for X, 99.69(4)% for Y_−π/2, and 99.62(5)% for X_π/2.

• Short-gate trade-off. Over gate durations from 50 to 200 ns, the TRIO pulse is limited at short times (t_f ≤ 90 ns) because its larger pulse area (3.093π versus a Gaussian π-pulse) demands a higher-amplitude envelope, raising leakage to roughly 10⁻³ compared with roughly 10⁻⁵ for the Gaussian pulse.

• Randomized-error statistics. With α and δ drawn from zero-mean Gaussians (σ_α = 0.2/3, σ_δ = 2/3 MHz, giving three-standard-deviation intervals α ∈ [−20, 20]% and δ ∈ [−2, 2] MHz), TRIO pulses show lower mean infidelities and distributions shifted toward lower infidelity for both X and H-type gates; the same offsets were applied to both pulse types for each realization.

• Energy-optimality. The optimized pulses have energies ℰ_X^opt = 3.461π and ℰ_H^opt = 5.633π, with pulse areas of 3.093π and 3.395π respectively.

• Projected improvement. Simulations predict TRIO gate errors below 10⁻⁶ for an anharmonicity of 1 GHz, as stated in the conclusion referencing the Supplemental Material.

Methodology in Plain English

The team treats the qubit as a two-level system driven by a microwave field with an amplitude and a detuning. Rather than guessing a pulse shape and simulating the result, they work backwards: they prescribe a smooth path for the qubit state on the Bloch sphere, then invert the equations of motion to find what amplitude and detuning would produce that path. Among all such paths that implement the target gate, they pick the one that minimizes pulse energy while also cancelling, to leading order in the error strengths, the effects of amplitude miscalibration and frequency offset. This is the RIO step.

The problem is that the inverse-engineered envelope tends to have abrupt, square-like features. So they add a time-rescaling step: they substitute an experimentally friendly smooth "hyper-Gaussian" envelope (a Gaussian raised to the fourth power) and solve for a nonlinear time reparametrization that maps it onto the original optimal envelope. Because the Schrödinger equation is invariant under such a reparametrization, the geometric trajectory — and therefore the robustness — is preserved. The matching detuning profile is then reconstructed and, being smooth, is compatible with DRAG corrections for leakage.

Experimentally, the reconstructed detuning is not applied through flux but encoded as a time-dependent phase on the microwave drive, which keeps the carrier frequency fixed; the real and imaginary parts of the complex envelope go to the I and Q channels. Robustness is tested by deliberately introducing errors: qubit-frequency detuning via the coupler's DC flux bias, and amplitude error by changing the pulse generator output. Gate fidelity and leakage are measured with interleaved randomized benchmarking, and the X gate is additionally characterized by quantum process tomography. Gaussian pulses with σ = t_f/6 serve as the baseline throughout.

Why This Matters

The work moves robust control pulses from theory into a real superconducting device and shows the robustness does not cost much in peak fidelity. That matters because calibration drift, crosstalk, and device-to-device variation are among the dominant practical obstacles to scaling superconducting processors: pulses that tolerate a wider error window reduce how often a large chip must be recalibrated.

Real-world applications (as implications of the platform):

• Fault-tolerant quantum computing built on error-corrected superconducting hardware, where gate errors must sit below correction thresholds.
• Large-scale superconducting processors, where calibration drift and crosstalk across many qubits make per-qubit pulse optimization expensive.
• Quantum sensing, which the authors explicitly name as a beneficiary, since precise control there is often limited by control imperfections and environmental noise.
• Other quantum-computing architectures, which the authors note could host the same complex-envelope IQ-type implementation (for example amplitude and phase quadratures of optical or Raman fields in atomic and trapped-ion systems).

Industry relevance: Energy-optimal, drift-tolerant pulses address two commercial pain points — calibration overhead and the cooling budget of large cryogenic systems — and the authors frame energy optimality as relevant to the ongoing debate over the energy costs of quantum technologies and the difficulty of cooling large superconducting processors.

Future Directions

• Shorter gates. Combining TRIO with higher-bandwidth waveform generation, active leakage mitigation, or high-order N-level DRAG is proposed to push gate durations below the 90 ns regime where the TRIO pulse currently suffers elevated leakage.

• Better materials and qubit types. The framework could be deployed in systems with larger anharmonicity or longer coherence times, including fluxonium, where the predicted errors below 10⁻⁶ at 1 GHz anharmonicity suggest substantial headroom.

• Extension beyond single-qubit control. The authors position TRIO as a foundation for robust control more broadly and point to other quantum-computing architectures and to quantum sensing as natural next targets.

• General open question. Whether the same inverse-optimal-plus-time-rescaling recipe can deliver comparable robustness for two-qubit gates on tunable couplers is not addressed here; the paper reports only single-qubit results on one qubit of a two-qubit device.

Target Audience

Quantum-control researchers and superconducting-qubit experimentalists who design pulse-level gates; graduate students and postdocs working on calibration, benchmarking (IRB, QPT), or leakage mitigation; and quantum-hardware engineers evaluating whether optimal-control waveforms are practical on real microwave hardware. Readers interested only in high-level quantum-computing trends will find the introduction and conclusion accessible, while the robustness analysis requires comfort with perturbation theory and Bloch-sphere trajectory engineering.

Authors’ abstract

Fault-tolerant quantum computing requires high-fidelity quantum gates that are resilient to control errors and device imperfections. Here, we develop energy-optimal control pulses for single-qubit gates that are simultaneously robust against frequency detuning and waveform inhomogeneities while also suppressing leakage outside the computational subspace. Combining the optimized pulse envelopes with an adaptive control scheme, we experimentally demonstrate optimal robust gates on a superconducting quantum processor. For a gate duration of 90ns, we achieve fidelities above 99.9% for the X gate and 99.6% for Hadamard-type gates. For the X gate, we maintain fidelities above 99.5% for simultaneous drive-amplitude variations up to 20% and qubit detuning up to 4 MHz. This robustness outperforms that of conventional Gaussian pulses by an order of magnitude, without much difference in peak gate fidelity. These results establish a route towards robust single-qubit control in large-scale quantum processors, advancing the development of fault-tolerant quantum computing.