We study the problem of optimally generating quantum gates in a logical subspace embedded in a larger Hilbert space, where the dynamics is also affected by unknown static imperfections. This general problem is widespread across various emergent quantum technology architectures. We derive the fidelity susceptibility in the computational subspace as a measure of robustness to perturbations, and define a cost function that quantifies leakage out of the subspace. We tackle both effects using a two-stage optimization where two cost functions are minimized in series. Specifically, we apply this framework to the generation of single-qubit gates in a superconducting transmon system, and find high-fidelity solutions robust to detuning and amplitude errors across various parameter regimes. We also show control pulses which maximize fidelity while minimizing leakage at all times during the evolution. However, finding control solutions that address both effects simultaneously is shown to be much more challenging, indicating the presence of a trade-off relation.
We study the problem of optimally generating quantum gates in a logical subspace embedded in a larger Hilbert space, where the dynamics is also affected by unknown static imperfections. This general problem is widespread across various emergent quantum technology architectures. We derive the fidelity susceptibility in the computational subspace as a measure of robustness to perturbations, and define a cost function that quantifies leakage out of the subspace. We tackle both effects using a two-stage optimization where two cost functions are minimized in series. Specifically, we apply this framework to the generation of single-qubit gates in a superconducting transmon system, and find high-fidelity solutions robust to detuning and amplitude errors across various parameter regimes. We also show control pulses which maximize fidelity while minimizing leakage at all times during the evolution. However, finding control solutions that address both effects simultaneously is shown to be much more challenging, indicating the presence of a trade-off relation.
In the companion paper (Part I), we developed an optimal-control framework to suppress intrinsic and decoherence-induced errors in controlled-Z (CZ) gates between directly coupled, frequency-tunable transmons. Here, we extend this framework to incorporate uncertainties in coupling strength, anharmonicities, and zero-flux qubit frequencies, enabling the design of high-fidelity gates robust against noise and parameter uncertainties. We classify these uncertainties by whether they perturb the off-diagonal coupling strengths or the diagonal detunings of the Hamiltonian, and we identify distinct suppression mechanisms for each class. Coupling errors are mitigated through dynamical decoupling induced by strong detuning modulation. For a 30-ns gate, this strategy reduces the coupling error to 6.21×10−7 and tolerates coupling-strength uncertainties up to 2π×4 MHz while keeping total gate infidelity below 10−3. Detuning errors are minimized near a symmetric-detuning operating point that balances the two dominant leakage-mediated interactions. A 24-ns gate operated near this point tolerates individual uncertainties of up to 2π×5 MHz in each relevant diagonal parameter under the same infidelity threshold. We further optimize gate performance under realistic conditions where intrinsic errors, decoherence, and parameter uncertainties coexist across six representative stages of progressively improving hardware performance. The resulting stage-specific controls achieve lower infidelity and greater robustness. Together with the companion study (Part I), this framework offers a practical approach to balance speed, fidelity, and robustness while reducing the need for exhaustive characterization and frequent recalibration in large-scale superconducting quantum processors.
High-fidelity quantum operations require the system dynamics to be strictly confined to the computational subspace. In practice, however, control fields inevitably couple to leakage levels, giving rise to quantum state leakage that significantly reduces the fidelity of the operation. To address this challenge, we propose a general strategy for actively suppressing leakage errors by applying small, static offsets to tunable system parameters. This approach systematically mitigates leakage's detrimental impact on quantum control, without modifying the original control framework or incurring additional time overhead. By avoiding the need for extra suppression pulses or complex optimization procedures altogether, it offers a streamlined solution for leakage compensation while remaining fully compatible with subsequent optimal control techniques. Numerical validation conducted on superconducting quantum circuits demonstrates effective leakage suppression, enabling high-fidelity single-qubit gates, precise control of two-qubit interactions, and perfect state transfer in multi-level systems. Moreover, when integrated with optimal control techniques, our approach also allows for the cooperative suppression of both leakage errors and residual crosstalk. Therefore, this work provides a feasible technical pathway toward the low error thresholds required for fault-tolerant quantum computation.
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.
Abstract Reaching high-speed, high-fidelity qubit operations requires precise control over the shape of the underlying pulses. For weakly anharmonic systems, such as superconducting transmon qubits, short gates lead to leakage to states outside of the computational subspace. Control pulses designed with open-loop optimal control may reduce such leakage. However, model inaccuracies can severely limit the usability of such pulses. We implemented a closed-loop optimization that simultaneously adapts all control parameters based on measurements of a cost function built from Clifford gates. We directly optimize the amplitude and phase of each sample point of the digitized control pulse. We thereby fully exploit the capabilities of the pulse generation electronics and create a 4.16 ns single-qubit pulse with 99.76 % fidelity and 0.044 % leakage. This is a sevenfold reduction of the leakage rate and a threefold reduction in standard errors of the best DRAG pulse we have calibrated at such short durations on the same system.
Superconducting transmon qubits are a promising platform for quantum computation, yet they face significant fidelity degradation due to connectivity noise, particularly in the intermediate coupling regime where noise levels are substantial. While prior works suggest that high fidelity requires operating in regimes with strongly suppressed noise, maintaining such conditions under practical experimental constraints remains challenging. To address this, we investigate quantum gate operations in fully connected transmon rings, examining both SWAP and general circuits. Our study reveals that fidelity can be significantly enhanced by tuning gate operation durations, with local maxima emerging even under strong noise conditions. These fidelity enhancements occur consistently across different qubit numbers and operation types, and for specific initial states -- particularly those with favorable symmetry or entanglement properties -- the achieved fidelities approach quantum error correction thresholds. Furthermore, we develop a supervised machine learning model that accurately predicts the optimal operation durations for new devices, enabling efficient optimization without extensive experimental simulations. These results provide a pathway toward robust quantum circuit design in noisy experimental environments.
Superconducting transmon qubits comprise one of the most promising platforms for quantum information processing due to their long coherence times and to their scalability into larger qubit networks. However, their weakly anharmonic spectrum leads to spectral crowding in multiqubit systems, making it challenging to implement fast, high-fidelity gates while avoiding leakage errors. To address this challenge, we use a protocol for speeding up waveforms by inducing phases to harmful transitions (SWIPHT) [S. E. Economou and E. Barnes, Phys. Rev. B 91, 161405(R) (2015)], which yields smooth, simple microwave pulses designed to suppress leakage without sacrificing gate speed through spectral selectivity. Here, we determine the parameter regimes in which SWIPHT is effective and demonstrate that in these regimes, it systematically produces two-qubit gate fidelities for cavity-coupled transmons in the range 99.6%--99.9% with gate times as fast as 23 ns. Our results are obtained from full numerical simulations that include current experimental levels of relaxation and dephasing. These high fidelities persist over a wide range of system parameters that encompass many current experimental setups and are insensitive to small parameter variations and pulse imperfections.