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Continuous Unitary Designs for Universally Robust Quantum Control

Unitary designs are unitary ensembles that emulate Haar-random unitary statistics. They provide a vital tool for studying quantum randomness and have found broad applications in quantum technologies. However, existing research has focused on discrete ensembles, despite that many physical processes, such as in quantum chaos, thermalization, and control, naturally involve continuous ensembles generated from continuous time-evolution. Here we initial the study of continuous unitary designs, addressing fundamental questions about their construction and practical utility. For single-qubit system, we construct explicit unitary 1-design paths from spherical 2-design curves and Hopf fibration theory. For arbitrary dimensions, we develop two systematic construction frameworks, one based on topological bundle theory of the unitary group and the other based on the Heisenberg-Weyl group. On the practical front, our unitary design paths provide analytical solutions to universally robust quantum control. Simulations show they outperform conventional pulse techniques in mitigating arbitrary unknown static noises, demonstrating immediate utility for quantum engineering. Extending unitary designs to the continuous domain not only introduces powerful geometric and topological tools that complement conventional combinatorial and group-theoretic methods, but also enhances experimental feasibility over discrete counterparts which usually involve instantaneous pulses. As an outlook, we anticipate that this work will pave the way for using continuous unitary designs to explore complex quantum dynamics and devise quantum information protocols.