Lie-Algebraic Bounds on Quantum Control Time via the Baker-Campbell-Hausdorff Formula

Abstract:The time required to implement a desired unitary operation is a central issue in quantum control, especially for many-body systems with limited control access. While controllability theory determines whether a target unitary is reachable in principle, it does not directly quantify the implementation time. Here we derive a Baker-Campbell-Hausdorff (BCH)-based inequality that connects these two questions at the operator level. Whenever a target unitary is implemented by the available Hamiltonians, an effective logarithmic generator can be chosen within the corresponding dynamical Lie algebra, and its normalized traceless Hilbert-Schmidt norm is bounded by the time integral of the same norm of the applied Hamiltonian. This induces a global-phase-insensitive distance on the reachable subgroup and yields a protocol-independent lower bound on the control time that explicitly respects Lie-algebraic restrictions. We compare the resulting bound with familiar quantum-speed-limit estimates and show that it refines a stabilizer-based control-time bound for an XY spin chain in the single-excitation subspace. Our result thus provides an algebraic link between controllability and quantitative bounds on unitary implementation time.

16 pages, 2 figures. Plus 19 pages of appendices

arXiv:2411.13155 [quant-ph]

(or arXiv:2411.13155v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2411.13155 Focus to learn more arXiv-issued DOI via DataCite

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