Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator Growth

Niklas Hörnedal1, Nicoletta Carabba1, Kazutaka Takahashi1,2, and Adolfo del Campo1,3

1Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg
2Department of Physics Engineering, Faculty of Engineering, Mie University, Mie 514–8507, Japan
3Donostia International Physics Center, E-20018 San Sebastián, Spain

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Quantum speed limits (QSLs) provide lower bounds on the minimum time required for a process to unfold by using a distance between quantum states and identifying the speed of evolution or an upper bound to it. We introduce a generalization of QSL to characterize the evolution of a general operator when conjugated by a unitary. The resulting operator QSL (OQSL) admits a geometric interpretation, is shown to be tight, and holds for operator flows induced by arbitrary unitaries, i.e., with time- or parameter-dependent generators. The derived OQSL is applied to the Wegner flow equations in Hamiltonian renormalization group theory and the operator growth quantified by the Krylov complexity.

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