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arXiv:2503.11204 · PRX Quantum

Realizing a Continuous Set of Two-Qubit Gates Parameterized by an Idle Time

Colin Scarato, Kilian Hanke, Ants Remm, Stefania Lazar, Nathan Lacroix, Dante Colao Zanuz, Alexander Flasby, Andreas J. Wallraff, Christoph Hellings

Published Aug 4, 2025• 6 citations
arXivHTMLPDF

Abstract

Continuous gate sets are a key ingredient for near-term quantum algorithms. Here, we demonstrate a hardware-efficient, continuous set of controlled arbitrary-phase ( CZ θ ) gates acting on flux-tunable transmon qubits. This implementation is robust to control pulse distortions on time scales longer than the duration of the gate, making it suitable for deep quantum circuits. Our calibration procedure makes it possible to parameterize the continuous gate set with a single control parameter, the idle time between the two rectangular halves of the net-zero control pulse. For calibration and characterization, we develop a leakage measurement based on coherent amplification, and a new cycle design for cross-entropy benchmarking. We demonstrate gate errors of 0.7 % and leakage of 4 × 10 − 4 across the entire gate set. This native gate set has the potential to reduce the depth and improve the performance of near-term quantum algorithms compared to decompositions into CZ π gates and single-qubit gates. Moreover, we expect the calibration and benchmarking methods to find further possible applications.

DOI
10.1103/h7cv-xgw2

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•Jan 1, 2020•59 citations
Continuous gate sets are a key ingredient for near-term quantum algorithms. Here, we demonstrate a hardware-efficient, continuous set of controlled arbitrary-phase ( CZ θ ) gates acting on flux-tunable transmon qubits. This implementation is robust to control pulse distortions on time scales longer than the duration of the gate, making it suitable for deep quantum circuits. Our calibration procedure makes it possible to parameterize the continuous gate set with a single control parameter, the idle time between the two rectangular halves of the net-zero control pulse. For calibration and characterization, we develop a leakage measurement based on coherent amplification, and a new cycle design for cross-entropy benchmarking. We demonstrate gate errors of 0.7 % and leakage of 4 × 10 − 4 across the entire gate set. This native gate set has the potential to reduce the depth and improve the performance of near-term quantum algorithms compared to decompositions into CZ π gates and single-qubit gates. Moreover, we expect the calibration and benchmarking methods to find further possible applications.
Quantum algorithms offer a dramatic speedup for computational problems in material science and chemistry. However, any near-term realizations of these algorithms will need to be optimized to fit within the finite resources offered by existing noisy hardware. Here, taking advantage of the adjustable coupling of gmon qubits, we demonstrate a continuous two-qubit gate set that can provide a threefold reduction in circuit depth as compared to a standard decomposition. We implement two gate families: an imaginary swap-like (iSWAP-like) gate to attain an arbitrary swap angle, θ, and a controlled-phase gate that generates an arbitrary conditional phase, ϕ. Using one of each of these gates, we can perform an arbitrary two-qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic simulation (fSim) gate set. We benchmark the fidelity of the iSWAP-like and controlled-phase gate families as well as 525 other fSim gates spread evenly across the entire fSim(θ,ϕ) parameter space, achieving a purity-limited average two-qubit Pauli error of 3.8×10^{-3} per fSim gate.
For superconducting quantum processors, stable high-fidelity two-qubit operations depend on precise flux control of the tunable coupler. However, the pulse distortion poses a significant challenge to the control precision. Current calibration methods, which often rely on microwave crosstalk or additional readout resonators for coupler excitation and readout, tend to be cumbersome and inefficient, especially when couplers only have flux control. Here, we introduce and experimentally validate a pulse calibration scheme that exploits the strong coupling between qubits and couplers, eliminating the need for extra coupler readout and excitation. Our method directly measures the short-time and long-time step responses of the coupler flux pulse transient, enabling us to apply predistortion to subsequent signals using fast Fourier transformation and deconvolution. This approach not only simplifies the calibration process, but also significantly improves the precision and stability of the flux control. We demonstrate the efficacy of our method through the implementation of diabatic controlled-ZZZ (cz) and iiiswap gates with fidelities of $99.61\mathrm{%}\ifmmode\pm\else\textpm\fi{}0.04\mathrm{%}$ and $99.82\mathrm{%}\ifmmode\pm\else\textpm\fi{}0.02\mathrm{%}$, respectively, as well as a series of diabatic controlled-phase gates with high fidelities characterized by cross-entropy benchmarking. The consistency and robustness of our technique are further validated by the reduction in pulse distortion and phase error observed across multilayer cz gates. These results underscore the potential of our calibration and predistortion method to enhance the performance of two-qubit gates in superconducting quantum processors.
Practical quantum advantage hinges on executing deep quantum circuits within the coherence limits of noisy intermediate-scale quantum processors. The absence of native, high-fidelity multi-qubit gates remains a major bottleneck, as their decomposition into single- and two-qubit gates leads to prohibitive depth and error overhead. Here, we propose a hardware-efficient protocol that directly implements a native controlled-controlled-Z (CCZ) gate in a tunable-coupler superconducting circuit. Our theoretical protocol activates a resonant three-qubit interaction via a time-frequency correlated virtual process, explicitly relying on the dynamic resonant exchange within the ∣101⟩↔∣020⟩|101\rangle \leftrightarrow |020\rangle∣101⟩↔∣020⟩ transition manifold. This approach is compatible with standard tunable-coupler architectures without requiring additional control resources. Through a systematic calibration workflow combining pulse shaping and active cancellation of residual phases, we demonstrate a gate fidelity exceeding 99\% within 165 ns165\,\mathrm{ns}165ns -- significantly outperforming decomposed sequences. Comprehensive error budgeting confirms that the gate performance remains robust against realistic experimental imperfections. Furthermore, we show that this scheme can be naturally extended to a continuous CCPhase(θ)\mathrm{CCPhase}(θ)CCPhase(θ) gate set. This work provides a direct, high-fidelity route to three-qubit entanglement, offering promising prospects for efficient execution of quantum algorithms on near-term superconducting hardware.
Ensemble nitrogen-vacancy (NV) centers are widely used for quantum sensing and magnetometry, where performance depends on the accurate implementation of microwave pulse sequences. In these systems, inhomogeneous broadening and short coherence times limit reliable multi-pulse control, while hardware imperfections and off-resonant excitation distort implemented rotations, making direct calibration of arbitrary gates tedious. To address this, we implement a control framework with a ZXZXZ gate decomposition which uses fixed-angle, phase-parameterized pulses, enabling universal single-qubit control from a single calibrated $ π/2$ primitive (Xπ/2X_{π/2}Xπ/2​) along with virtual Z operations. We compare a standard pulse with a composite pulse designed for high-power operation in a regime where the Rabi frequency is comparable to the hyperfine splitting, leading to non-uniform evolution across hyperfine levels. Using pre-determined combinations of π/2π/2π/2 pulse sequences, we observe that standard pulses rapidly deviate from ideal behavior under repeated application, while the composite pulse suppresses this error accumulation by more than 70%\%% for longer sequences despite its longer duration. In contrast, randomized benchmarking yields similar decay factors for both implementations, highlighting its limited sensitivity to coherent control errors. These results establish a practical route to robust control in short-coherence ensemble systems and emphasize the importance of pulse-level characterization.
Hybrid superconducting architectures combining different types of qubits offer a promising platform for exploiting their complementary advantages, yet high-fidelity entangling gates remain challenging because of strong nonlinearities and residual qubit-qubit interactions. Here, we propose a high-fidelity controlled-Z (CZ) gate for a hybrid circuit comprising a fluxonium qubit, a fixed-frequency transmon qubit, and a flux-tunable transmon coupler. By modulating only the external magnetic flux applied to the coupler, the qubit-qubit interaction is dynamically engineered for conditional-phase accumulation while the residual interaction is suppressed at idle, mitigating spectator-induced errors. We employ a low-dimensional Fourier-cosine pulse parameterization and a physically motivated cost function to independently suppress conditional-phase errors and leakage from the computational subspace. Numerical simulations demonstrate that a microwave-free CZ gate can be realized within 25ns25 \mathrm{ns}25ns, with an average gate fidelity exceeding 99.99%99.99\%99.99% and leakage below 10−510^{-5}10−5. Using experimentally relevant superconducting-qubit parameters and accounting for decoherence, the proposed scheme maintains a CZ-gate fidelity of approximately 99.9%99.9\%99.9%. We further extend the analysis to larger coupled architectures and find that the CZ-gate infidelity remains below 10−410^{-4}10−4 in the presence of spectator qubits. These results establish single-parameter flux control as a simple and robust approach for realizing high-fidelity entangling gates in heterogeneous superconducting quantum architectures.
Variational quantum algorithms are believed to be promising for solving computationally hard problems on noisy intermediate-scale quantum (NISQ) systems. Gaining computational power from these algorithms critically relies on the mitigation of errors during their execution, which for coherence-limited operations is achievable by reducing the gate count. Here, we demonstrate an improvement of up to a factor of 3 in algorithmic performance for the quantum approximate optimization algorithm (QAOA) as measured by the success probability, by implementing a continuous hardware-efficient gate set using superconducting quantum circuits. This gate set allows us to perform the phase separation step in QAOA with a single physical gate for each pair of qubits instead of decomposing it into two CZ gates and single-qubit gates. With this reduced number of physical gates, which scales with the number of layers employed in the algorithm, we experimentally investigate the circuit-depth-dependent performance of QAOA applied to exact-cover problem instances mapped onto three and seven qubits, using up to a total of 399 operations and up to nine layers. Our results demonstrate that the use of continuous gate sets may be a key component in extending the impact of near-term quantum computers.