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A fundamental problem in quantum engineering is determining the lowest timerequired to ensure that all possible unitaries can be generated with the toolsavailable, which is one of a number of possible quantum speed limits. Weexamine this problem from the perspective of quantum control, where the systemof interest is described by a drift Hamiltonian and set of controlHamiltonians. Our approach uses a combination of Lie algebra theory, Lie groupsand differential geometry, and formulates the problem in terms of geodesics ona differentiable manifold. We provide explicit lower bounds on the quantumspeed limit for the case of an arbitrary drift, requiring only that the controlHamiltonians generate a topologically closed subgroup of the full unitarygroup, and formulate criteria as to when our expression for the speed limit isexact and not merely a lower bound. These analytic results are then tested andconfirmed using a numerical optimization scheme. Finally we extend the analysisto find a lower bound on the quantum speed limit in the common case where thesystem is described by a drift Hamiltonian and a single control Hamiltonian.

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