  A basic question in the theory of fault-tolerant quantum computation is to
understand the fundamental resource costs for performing a universal logical
set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits
encoded with constant space overhead (i.e. finite encoding rate) in the limit
of arbitrarily large code distance $d$ through the use of topological codes
associated to triangulations of hyperbolic surfaces. We introduce explicit
protocols to demonstrate how Dehn twists of the hyperbolic surface can be
implemented on the code through constant depth unitary circuits, without
increasing the space overhead. The circuit for a given Dehn twist consists of a
permutation of physical qubits, followed by a constant depth local unitary
circuit, where locality here is defined with respect to a hyperbolic metric
that defines the code. Applying our results to the hyperbolic Fibonacci
Turaev-Viro code implies the possibility of applying universal logical gate
sets on encoded qubits through constant depth unitary circuits and with
constant space overhead. Our circuits are inherently protected from errors as
they map local operators to local operators while changing the size of their
support by at most a constant factor; in the presence of noisy syndrome
measurements, our results suggest the possibility of universal fault tolerant
quantum computation with constant space overhead and time overhead of
$\mathcal{O}(d/\log d)$. For quantum circuits that allow parallel gate
operations, this yields the optimal scaling of space-time overhead known to
date.
