Saturday, 19 August 2017

The Geometry of Speed Limiting Resources

Benjamin Russell, Susan Stepney.
The Geometry of Speed Limiting Resources in Physical Models of Computation.
International Journal of Foundations of Computer Science 28(4):321-333, 2017.

This is the latest paper in our series on using geometrical approaches to determining speed limits for quantum operations: it takes time to change a quantum state, which will limit the speed of quantum computers.  The series started with a generalisation of the Zeppelin navigation problem, and continued with a further generalisation allowing us to use the word “brachistochrone” in the title.  The current paper is a further generalisation still, to a wider class of systems.

We study the maximum speed of quantum computation and how it is affected by limitations on physical resources. We show how the resulting concepts generalize to a broader class of physical models of computation within dynamical systems and introduce a specific algebraic structure representing these speed limits. We derive a family of quantum speed limit results in resource-constrained quantum systems with pure states and a finite dimensional state space, by using a geometric method based on right invariant action functionals on SU(N). We show that when the action functional is bi-invariant, the minimum time for implementing any quantum gate using a potentially time-dependent Hamiltonian is equal to the minimum time when using a constant Hamiltonian, thus constant Hamiltonians are time optimal for these constraints. We give an explicit formula for the time in these cases, in terms of the resource constraint. We show how our method produces a rich family of speed limit results, of which the generalized Margolus–Levitin theorem and the Mandelstam–Tamm inequality are special cases. We discuss the broader context of geometric approaches to speed limits in physical computation, including the way geometric approaches to quantum speed limits are a model for physical speed limits to computation arising from a limited resource.

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