APPLICATIONS OF FINSLER GEOMETRY TO SPEED LIMITS TO QUANTUM INFORMATION PROCESSING
International Journal of Foundations of Computer Science
We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Here, we discuss speed limits in the context of quantum computing. We review some relevant parts of the theory of Finsler metrics on Lie groups and homogeneous spaces such as the special unitary groups and complex projective spaces. We show how these constructions can be applied to
... alysing the limit to the speed of quantum information processing operations in constrained quantum systems with finite dimensional Hilbert spaces of states. We demonstrate the approach applied to a spin chain system. The Margolus-Levitin (ML) bound  indicates a limit to the speed of dynamical evolution of any quantum system with a time-independent Hamiltonian in terms of the system's energy expectation. This and other such speed limit bounds have an interpretation in terms of the maximum information processing rate of any quantum system  . This bound complements the Mandelstam-Tamm inequality , a bound to the speed of dynamical evolution of a quantum system in terms of the energy uncertainty. These bounds have been combined into the 'unified bound '  which is, in a sense, tight for all systems. Bounds are also know for the optimal time in which a quantum gate can be implemented  . There is a geometric derivation of the ML bound  . In the application of quantum optimal control to quantum computing, a timedependent Hamiltonian is more common  , and so a more complete analysis of the limit to the speed of quantum computers needs to take into account the time dependence of the Hamiltonian. A notable result analogous to the ML bound, applicable to time dependent systems in the adiabatic regime, can be found in  .