The Geometry of Speed Limiting Resources in Physical Models of Computation

TL;DR

This paper explores the fundamental limits on the speed of quantum computation imposed by physical resource constraints, using geometric methods to derive new and generalized quantum speed limit results within dynamical systems.

Contribution

It introduces a geometric framework for understanding speed limits in physical models of computation, deriving explicit formulas and generalizations of known quantum speed limits.

Findings

Constant Hamiltonians are time optimal under bi-invariant action functionals.

Derived a family of quantum speed limit results including generalizations of Margolus--Levitin and Mandelstam--Tamm inequalities.

Provided explicit formulas for minimum implementation times based on resource constraints.

Abstract

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.