128 research outputs found
Topological Quantum Error Correction with Optimal Encoding Rate
We prove the existence of topological quantum error correcting codes with
encoding rates asymptotically approaching the maximum possible value.
Explicit constructions of these topological codes are presented using surfaces
of arbitrary genus. We find a class of regular toric codes that are optimal.
For physical implementations, we present planar topological codes.Comment: REVTEX4 file, 5 figure
Topological Subsystem Codes
We introduce a family of 2D topological subsystem quantum error-correcting
codes. The gauge group is generated by 2-local Pauli operators, so that 2-local
measurements are enough to recover the error syndrome. We study the
computational power of code deformation in these codes, and show that
boundaries cannot be introduced in the usual way. In addition, we give a
general mapping connecting suitable classical statistical mechanical models to
optimal error correction in subsystem stabilizer codes that suffer from
depolarizing noise.Comment: 16 pages, 11 figures, explanations added, typos correcte
Universal topological phase of 2D stabilizer codes
Two topological phases are equivalent if they are connected by a local
unitary transformation. In this sense, classifying topological phases amounts
to classifying long-range entanglement patterns. We show that all 2D
topological stabilizer codes are equivalent to several copies of one universal
phase: Kitaev's topological code. Error correction benefits from the
corresponding local mappings.Comment: 4 pages, 3 figure
Entanglement Distillation Protocols and Number Theory
We show that the analysis of entanglement distillation protocols for qudits
of arbitrary dimension benefits from applying basic concepts from number
theory, since the set \zdn associated to Bell diagonal states is a module
rather than a vector space. We find that a partition of \zdn into divisor
classes characterizes the invariant properties of mixed Bell diagonal states
under local permutations. We construct a very general class of recursion
protocols by means of unitary operations implementing these local permutations.
We study these distillation protocols depending on whether we use twirling
operations in the intermediate steps or not, and we study them both
analitically and numerically with Monte Carlo methods. In the absence of
twirling operations, we construct extensions of the quantum privacy algorithms
valid for secure communications with qudits of any dimension . When is a
prime number, we show that distillation protocols are optimal both
qualitatively and quantitatively.Comment: REVTEX4 file, 7 color figures, 2 table
Error Threshold for Color Codes and Random 3-Body Ising Models
We study the error threshold of color codes, a class of topological quantum
codes that allow a direct implementation of quantum Clifford gates suitable for
entanglement distillation, teleportation and fault-tolerant quantum
computation. We map the error-correction process onto a statistical mechanical
random 3-body Ising model and study its phase diagram via Monte Carlo
simulations. The obtained error threshold of p_c = 0.109(2) is very close to
that of Kitaev's toric code, showing that enhanced computational capabilities
does not necessarily imply lower resistance to noise.Comment: 4 pages, 3 figures, 1 tabl
A note on the minimum distance of quantum LDPC codes
We provide a new lower bound on the minimum distance of a family of quantum
LDPC codes based on Cayley graphs proposed by MacKay, Mitchison and
Shokrollahi. Our bound is exponential, improving on the quadratic bound of
Couvreur, Delfosse and Z\'emor. This result is obtained by examining a family
of subsets of the hypercube which locally satisfy some parity conditions
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
Statistical Mechanical Models and Topological Color Codes
We find that the overlapping of a topological quantum color code state,
representing a quantum memory, with a factorized state of qubits can be written
as the partition function of a 3-body classical Ising model on triangular or
Union Jack lattices. This mapping allows us to test that different
computational capabilities of color codes correspond to qualitatively different
universality classes of their associated classical spin models. By generalizing
these statistical mechanical models for arbitrary inhomogeneous and complex
couplings, it is possible to study a measurement-based quantum computation with
a color code state and we find that their classical simulatability remains an
open problem. We complement the meaurement-based computation with the
construction of a cluster state that yields the topological color code and this
also gives the possibility to represent statistical models with external
magnetic fields.Comment: Revtex4, color figures, submitted for publicatio
Structure of 2D Topological Stabilizer Codes
We provide a detailed study of the general structure of two-dimensional
topological stabilizer quantum error correcting codes, including subsystem
codes. Under the sole assumption of translational invariance, we show that all
such codes can be understood in terms of the homology of string operators that
carry a certain topological charge. In the case of subspace codes, we prove
that two codes are equivalent under a suitable set of local transformations if
and only they have equivalent topological charges. Our approach emphasizes
local properties of the codes over global ones.Comment: 54 pages, 11 figures, version accepted in journal, improved
presentation and result
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