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

Understand

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.