591 research outputs found
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Topological arbiters
This paper initiates the study of topological arbiters, a concept rooted in
Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological
arbiter associates a value 0 or 1 to codimension zero submanifolds of W,
subject to natural topological and duality axioms. For example, there is a
unique arbiter on , which reports the location of the essential 1-cycle.
In contrast, we show that there exists an uncountable collection of topological
arbiters in dimension 4. Families of arbiters, not induced by homology, are
also shown to exist in higher dimensions. The technical ingredients underlying
the four dimensional results are secondary obstructions to generalized
link-slicing problems. For classical links in the 3-sphere the construction
relies on the existence of nilpotent embedding obstructions in dimension 4,
reflected in particular by the Milnor group. In higher dimensions novel
arbiters are produced using nontrivial squares in stable homotopy theory.
The concept of "topological arbiter" derives from percolation and from
4-dimensional surgery. It is not the purpose of this paper to advance either of
these subjects, but rather to study the concept for its own sake. However in
appendices we give both an application to percolation, and the current
understanding of the relationship between arbiters and surgery. An appendix
also introduces a more general notion of a multi-arbiter. Properties and
applications are discussed, including a construction of non-homological
multi-arbiters.Comment: v3: A minor reorganization of the pape
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