224 research outputs found
The geometry of quantum stabiliser codes
The aim of this project is to bring together quantum error-correcting codes theory and the study of finite geometries. A quantum code is used to protect quantum information from errors that may occur due to quantum decoherence. We give a geometric interpretation of the codes as sets of lines in certain finite projective spaces. We exploit the geometric aspect of codes to rewrite proofs in a more intuitive way and explore their properties through visualization. Some examples of stabiliser codes and their associated quantum sets of lines are presented. We also discuss how to build nonadditive codes as the union of stabiliser codes. Finite geometry has proved to be a powerful tool to work on quantum error-correcting codes. Some of its applications include finding new codes or proving the non-existence of codes with certain parameters
Codeword stabilized quantum codes: algorithm and structure
The codeword stabilized ("CWS") quantum codes formalism presents a unifying
approach to both additive and nonadditive quantum error-correcting codes
(arXiv:0708.1021). This formalism reduces the problem of constructing such
quantum codes to finding a binary classical code correcting an error pattern
induced by a graph state. Finding such a classical code can be very difficult.
Here, we consider an algorithm which maps the search for CWS codes to a problem
of identifying maximum cliques in a graph. While solving this problem is in
general very hard, we prove three structure theorems which reduce the search
space, specifying certain admissible and optimal ((n,K,d)) additive codes. In
particular, we find there does not exist any ((7,3,3)) CWS code though the
linear programming bound does not rule it out. The complexity of the CWS search
algorithm is compared with the contrasting method introduced by Aggarwal and
Calderbank (arXiv:cs/0610159).Comment: 11 pages, 1 figur
Approximate quantum error correction for generalized amplitude damping errors
We present analytic estimates of the performances of various approximate
quantum error correction schemes for the generalized amplitude damping (GAD)
qubit channel. Specifically, we consider both stabilizer and nonadditive
quantum codes. The performance of such error-correcting schemes is quantified
by means of the entanglement fidelity as a function of the damping probability
and the non-zero environmental temperature. The recovery scheme employed
throughout our work applies, in principle, to arbitrary quantum codes and is
the analogue of the perfect Knill-Laflamme recovery scheme adapted to the
approximate quantum error correction framework for the GAD error model. We also
analytically recover and/or clarify some previously known numerical results in
the limiting case of vanishing temperature of the environment, the well-known
traditional amplitude damping channel. In addition, our study suggests that
degenerate stabilizer codes and self-complementary nonadditive codes are
especially suitable for the error correction of the GAD noise model. Finally,
comparing the properly normalized entanglement fidelities of the best
performant stabilizer and nonadditive codes characterized by the same length,
we show that nonadditive codes outperform stabilizer codes not only in terms of
encoded dimension but also in terms of entanglement fidelity.Comment: 44 pages, 8 figures, improved v
Codeword Stabilized Quantum Codes for Asymmetric Channels
We discuss a method to adapt the codeword stabilized (CWS) quantum code
framework to the problem of finding asymmetric quantum codes. We focus on the
corresponding Pauli error models for amplitude damping noise and phase damping
noise. In particular, we look at codes for Pauli error models that correct one
or two amplitude damping errors. Applying local Clifford operations on graph
states, we are able to exhaustively search for all possible codes up to length
. With a similar method, we also look at codes for the Pauli error model
that detect a single amplitude error and detect multiple phase damping errors.
Many new codes with good parameters are found, including nonadditive codes and
degenerate codes.Comment: 5 page
Codeword Stabilized Quantum Codes
We present a unifying approach to quantum error correcting code design that
encompasses additive (stabilizer) codes, as well as all known examples of
nonadditive codes with good parameters. We use this framework to generate new
codes with superior parameters to any previously known. In particular, we find
((10,18,3)) and ((10,20,3)) codes. We also show how to construct encoding
circuits for all codes within our framework.Comment: 5 pages, 1 eps figure, ((11,48,3)) code removed, encoding circuits
added, typos corrected in codewords and elsewher
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
Quantum Error Correcting Codes Using Qudit Graph States
Graph states are generalized from qubits to collections of qudits of
arbitrary dimension , and simple graphical methods are used to construct
both additive and nonadditive quantum error correcting codes. Codes of distance
2 saturating the quantum Singleton bound for arbitrarily large and are
constructed using simple graphs, except when is odd and is even.
Computer searches have produced a number of codes with distances 3 and 4, some
previously known and some new. The concept of a stabilizer is extended to
general , and shown to provide a dual representation of an additive graph
code.Comment: Version 4 is almost exactly the same as the published version in
Phys. Rev.
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