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### 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

#### 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.

journal doi: 10.1142/S0129054117500204

@article(Russell-2017,
author = "Benjamin Russell and Susan Stepney",
title = "The Geometry of Speed Limiting Resources in Physical Models of Computation",
journal = "International Journal of Foundations of Computer Science",
volume = 28,
issue = 4,
pages = "321-333",
doi = "10.1142/S0129054117500204",
year = 2017
)