arXiv:1908.09253v1 [quant-ph] 25 Aug 2019Holographic Code Rate
Noah Bray-Ali∗
Department of Physics, California State University, Dominguez Hills, California 90747 USA
David Chester
Department of Physics and Astronomy, University of California, Los Angeles, California 90095 USA and
Quantum Gravity Research, Los Angeles, California 90290 USA
Dugan Hammock, Marcelo M. Amaral, and Klee Irwin
Quantum Gravity Research, Los Angeles, California 90290 USA
Michael F. Rios
Dyonica, ICMQG, Los Angeles, California 90032 USA
(Dated: August 27, 2019)
Holographic codes grown with perfect tensors on regular hyperbolic tessellations using an inflation
rule protect quantum information stored in the bulk from errors on the boundary provided the code
rate is less than one. Hyperbolic geometry bounds the holographic code rate and guarantees quantum
error correction for codes grown with any inflation rule on all regular hyperbolic tessellations in a
class whose size grows exponentially with the rank of the perfect tensors for rank five and higher.
For the tile completion inflation rule, holographic triangle codes have code rate more than one but
all others perform quantum error correction.
I.
INTRODUCTION
Holographic quantum error-correcting codes[1, 2]
merge quasi-crystals[3] and hyperbolic geometry[4, 5]
with quantum information[6–14] and holography[15–23].
One places rank-(p+ 1) perfect tensors Ta1a2...apap+1 on
the p-sided tiles of a tessellation of the hyperbolic plane
and contracts the tensors along the edges where tiles
meet: this leaves a single “bulk” index uncontracted for
each tile[1]. Starting from some simply connected set of
seed tiles, we grow the holographic code, layer by layer,
using an inflation rule[3].
The physical degrees of freedom of the code live on
the boundary of the growing tile set on the quasi-crystal
formed by the dangling edges of the tiles of the last
layer[1]. The logical degrees of freedom of the code live in
the bulk of the tile set on the tiles themselves. The per-
fect tensors map the physical H