Exact and lower bounds for the quantum speed limit in finite dimensional systems

Abstract

A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated with the tools available, which is one of a number of possible quantum speed limits. We examine this problem from the perspective of quantum control, where the system of interest is described by a drift Hamiltonian and set of control Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups and differential geometry, and formulates the problem in terms of geodesics on a differentiable manifold. We provide explicit lower bounds on the quantum speed limit for the case of an arbitrary drift, requiring only that the control Hamiltonians generate a topologically closed subgroup of the full unitary group, and formulate criteria as to when our expression for the speed limit is exact and not merely a lower bound. These analytic results are then tested and confirmed using a numerical optimization scheme. Finally we extend the analysis to find a lower bound on the quantum speed limit in the common case where the system is described by a drift Hamiltonian and a single control Hamiltonian.

13 pages

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• Quantum speed limit for physical processes
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Cited by in corpus (3)

• Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics
• A Unifying Quantum Speed Limit For Time-Independent Hamiltonian Evolution
• Tighter Lower Bounds on Quantum Annealing Times