Unit Fidelity Entangling Gates Achieved Via Continuous Dynamical Decoupling and Optimal Control

The creation of high-fidelity quantum gates remains a central challenge in the development of superconducting quantum computers. Adonai Hilário da Silva, Octávio da Motta, and Leonardo K. Castelano, from the Sao Carlos Institute of Physics and the Universidade Federal de São Carlos, present a novel approach combining continuous dynamical decoupling with variational minimal-energy optimal control to address this issue. Their research demonstrates a unified scheme for generating tunable two-qubit gates while actively suppressing noise and calibration errors, leading to a remarkably stable system. By minimising gate infidelity through a variational geodesic optimisation process, the team achieves virtually perfect fidelity and robustness for gates such as CZ and CX, representing a significant step towards practical, noise-resilient quantum computation. This methodology, further supported by the work of Reginaldo de Jesus Napolitano, establishes a promising pathway for designing reliable entangling gates in superconducting platforms.

The research combines continuous dynamical decoupling (CDD) with variational minimal-energy optimal control to suppress residual couplings, calibration drift, and quasistatic noise, resulting in a stable effective Hamiltonian that preserves the designed ZZ interaction crucial for tunable couplers. This innovative approach establishes a stable manifold within which smooth, low-energy single-qubit control functions are calculated, directly minimizing gate infidelity through a variational geodesic optimization process. The team successfully applied this methodology to create CZ, CX, and generic entangling gates, showcasing experimentally realistic control fields and establishing a practical, noise-resilient scheme for designing superconducting entangling gates.

The study reveals a two-stage process where continuous dynamical decoupling preemptively eliminates low-frequency noise and spurious static terms in the Hamiltonian, effectively averaging out undesired perturbations during gate execution. Following this initial suppression, optimized control fields are introduced to drive the effective two-qubit gate, steering the evolution along a geodesic trajectory in SU(4) generated by a control Hamiltonian. This trajectory is determined by a single parametric matrix, Λ(0), which defines the entire time dependence of the control fields through a nonlinear Schrödinger equation. By optimizing gate fidelity, the researchers implemented a second Lagrange multiplier matrix, Γ(t), propagated backward in time, yielding an efficient boundary value optimization over Λ(0) that circumvents the need for neural networks or complex pulse shaping techniques.

Experiments show high-fidelity two-qubit entangling gates achieved through this unified control scheme, demonstrating robustness against coherent noise and crosstalk. Unlike conventional multi-layered approaches that address error sources separately, this methodology achieves both robustness and control effectiveness through a single optimization process. Simulations were conducted modeling superconducting transmons as ideal two-state qubits, though the variational framework is designed to be extensible to higher-dimensional control spaces, paving the way for future investigations into leakage suppression during fast gate operations. The team intends to expand the current SU(4) model to SU(9) to further refine control and fidelity.

This work establishes CDD-enhanced variational geometric optimal control as a significant advancement in quantum gate design, offering a systematic and unified approach to overcome persistent two-qubit vulnerabilities. By addressing low-frequency flux fluctuations, calibration drift, and spurious couplings within a single framework, the research provides a pathway towards scalable quantum computing. The ability to achieve virtually unit fidelity with experimentally realistic control fields positions this methodology as a promising solution for building robust and reliable quantum processors capable of surpassing the limitations of current noisy intermediate-scale quantum (NISQ) technology.

CDD and Variational Control for Entangling Gates

The research team pioneered a unified methodology for generating high-fidelity entangling gates, combining continuous dynamical decoupling (CDD) with variational minimal-energy optimal control to address limitations in current quantum computing platforms. The study began by employing CDD to actively suppress residual couplings, calibration drift, and quasistatic noise, establishing a stable effective Hamiltonian that preserves the desired ZZ interaction crucial for tunable couplers. This initial stage effectively mitigates low-frequency noise and spurious static terms, creating a robust foundation for subsequent gate operations. Following the CDD phase, scientists calculated smooth, low-energy single-qubit control functions using a variational geodesic optimization process, directly minimizing gate infidelity.

This innovative approach operates within a stable SU(4) manifold, leveraging a control Hamiltonian projected from an auxiliary co-state matrix, Λ(0), to determine the temporal evolution of control fields. A second Lagrange multiplier matrix, Γ(t), was propagated backward in time, utilizing a second-variation principle to efficiently optimize Λ(0) without relying on neural networks or complex pulse shaping techniques. Experiments focused on implementing CZ, CX, and generic entangling gates, achieving virtually unit fidelity and demonstrating robustness even under restricted single-qubit action with realistic control fields. The team modeled superconducting transmons as ideal two-state qubits, initially disregarding leakage, but the variational framework is designed for extension to higher-dimensional control spaces, including a planned expansion to SU(9) to address leakage suppression in future work.

The physical model begins with a two-qubit Hamiltonian, assuming a free evolution with a frequency of 5GHz, and an intentionally tuned ZZ coupling fixed at 82.5MHz. This CDD-enhanced variational geometric optimal control scheme represents a significant advancement, offering a practical and noise-resilient solution for designing superconducting entangling gates and circumventing the need for multiple layers of error correction typically required in current implementations. The unified optimization process, focusing on Λ(0), streamlines control and enhances both robustness and effectiveness, paving the way for more scalable and reliable quantum computation.

High-Fidelity Entangling Gates via Optimal Control

Scientists achieved virtually unit fidelity in generating entangling gates using a novel scheme combining continuous dynamical decoupling (CDD) with variational geometric optimal control. The research demonstrates a unified approach to suppress residual couplings, calibration drift, and quasistatic noise, establishing a stable effective Hamiltonian crucial for producing tunable couplers. Experiments reveal that this stable manifold allows for the calculation of smooth, low-energy single-qubit control functions, directly minimizing gate infidelity during operation. This methodology was successfully applied to CZ, CX, and generic entangling gates, yielding remarkably high performance metrics under restricted single-qubit action with experimentally realistic control fields.

The team measured a ZZ coupling intentionally tuned to 82.5MHz, operating with a qubit frequency of 5GHz, and transformed the system into a rotating frame to isolate relevant entanglement and control terms. This transformation eliminates fast oscillations, enabling an effective description where ideal qubits are gapless and only interactions and control fields contribute to the dynamics. Results demonstrate the successful implementation of a general native Hamiltonian, expressed as a sum of interactions between Pauli matrices, within the SU(4) manifold. Calculations confirm that by restricting analysis to SU(4), the Hamiltonian remains traceless, simplifying gate design and analysis.

Further work involved calculating smooth low-energy single-qubit control functions using a variational geodesic optimization process, minimizing gate infidelity through a nonlinear Schrödinger equation. The breakthrough delivers a second-variation principle, propagating a Lagrange multiplier matrix backward in time to efficiently optimize gate fidelity without relying on neural networks or complex pulse shaping. Simulations show robustness against coherent noise and crosstalk, achieving control effectiveness through a single optimization process. Measurements confirm that this unified control scheme surpasses multi-layered approaches, which address error sources separately, by integrating robustness and control into a single, streamlined process. While initial modeling focused on ideal two-state qubits, the variational framework is readily extensible to higher-dimensional control spaces, with future investigations planned to extend the model to SU(9) to further suppress leakage during fast gate operations. This research establishes CDD-enhanced variational geometric optimal control as a practical and noise-resilient scheme for designing superconducting entangling gates.