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

TL;DR

This paper introduces a new geometric quantum speed limit for operators, applicable to various unitary evolutions, and demonstrates its relevance in Hamiltonian renormalization and operator growth analysis.

Contribution

It generalizes quantum speed limits to operators, providing a tight, geometric bound applicable to arbitrary unitary-induced operator flows.

Findings

The operator QSL (OQSL) is tight and geometrically interpretable.

OQSL applies to Wegner flow equations in Hamiltonian renormalization.

OQSL quantifies operator growth via Krylov complexity.

Abstract

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.