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

• URL: http://arxiv.org/abs/2304.06617v1
• Date: Thu, 13 Apr 2023 15:37:39 GMT
• Title: Exact and lower bounds for the quantum speed limit in finite dimensional systems
• Authors: Mattias T. Johnsson, Lauritz van Luijk, Daniel Burgarth
• Abstract summary: A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated. We use Lie algebra theory, Lie groups and differential geometry to formulate the problem in terms of geodesics on a differentiable manifold. We 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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