19,369 research outputs found
Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study
We study the resources needed to construct topological 2D stabilizer codes as
a way to estimate in part their efficiency and this leads us to perform a
comparative study of surface codes and color codes. This study clarifies the
similarities and differences between these two types of stabilizer codes. We
compute the error correcting rate for surface codes and color
codes in several instances. On the torus, typical values are and
, but we find that the optimal values are and . For
planar codes, a typical value is , while we find that the optimal values
are and . In general, a color code encodes twice as much
logical qubits as a surface code does.Comment: revtex, 6 pages, 7 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
Topological Computation without Braiding
We show that universal quantum computation can be performed within the ground
state of a topologically ordered quantum system, which is a naturally protected
quantum memory. In particular, we show how this can be achieved using brane-net
condensates in 3-colexes. The universal set of gates is implemented without
selective addressing of physical qubits and, being fully topologically
protected, it does not rely on quasiparticle excitations or their braiding.Comment: revtex4, 4 pages, 4 figure
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 color codes on Union Jack lattices: A stable implementation of the whole Clifford group
We study the error threshold of topological color codes on Union Jack
lattices that allow for the full implementation of the whole Clifford group of
quantum gates. After mapping the error-correction process onto a statistical
mechanical random 3-body Ising model on a Union Jack lattice, we compute its
phase diagram in the temperature-disorder plane using Monte Carlo simulations.
Surprisingly, topological color codes on Union Jack lattices have similar error
stability than color codes on triangular lattices, as well as the Kitaev toric
code. The enhanced computational capabilities of the topological color codes on
Union Jack lattices with respect to triangular lattices and the toric code
demonstrate the inherent robustness of this implementation.Comment: 8 pages, 4 figures, 1 tabl
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