359 research outputs found
Quantum Convolutional BCH Codes
Quantum convolutional codes can be used to protect a sequence of qubits of
arbitrary length against decoherence. We introduce two new families of quantum
convolutional codes. Our construction is based on an algebraic method which
allows to construct classical convolutional codes from block codes, in
particular BCH codes. These codes have the property that they contain their
Euclidean, respectively Hermitian, dual codes. Hence, they can be used to
define quantum convolutional codes by the stabilizer code construction. We
compute BCH-like bounds on the free distances which can be controlled as in the
case of block codes, and establish that the codes have non-catastrophic
encoders.Comment: 4 pages, minor changes, accepted for publication at the 10th Canadian
Workshop on Information Theory (CWIT'07
Quantum error control codes
It is conjectured that quantum computers are able to solve certain problems more
quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm
is able to factor large integers in polynomial time on a quantum computer.
A quantum computer exploits the rules of quantum mechanics to speed up computations.
However, it is a formidable task to build a quantum computer, since the
quantum mechanical systems storing the information unavoidably interact with their
environment. Therefore, one has to mitigate the resulting noise and decoherence
effects to avoid computational errors.
In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the
fundamental theory and necessary background of quantum codes and construct many
families of quantum block and convolutional codes over finite fields, in addition to
families of subsystem codes. This dissertation is organized into three parts:
Quantum Block Codes. After introducing the theory of quantum block codes, I
establish conditions when BCH codes are self-orthogonal (or dual-containing)
with respect to Euclidean and Hermitian inner products. In particular, I derive
two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum
codes, as well as families of quantum codes derived from projective geometries.
Subsystem Codes. Subsystem codes form a new class of quantum codes in which
the underlying classical codes do not need to be self-orthogonal. I give an
introduction to subsystem codes and present several methods for subsystem
code constructions. I derive families of subsystem codes from classical BCH and
RS codes and establish a family of optimal MDS subsystem codes. I establish
propagation rules of subsystem codes and construct tables of upper and lower
bounds on subsystem code parameters.
Quantum Convolutional Codes. Quantum convolutional codes are particularly
well-suited for communication applications. I develop the theory of quantum
convolutional codes and give families of quantum convolutional codes based
on RS codes. Furthermore, I establish a bound on the code parameters of
quantum convolutional codes - the generalized Singleton bound. I develop a
general framework for deriving convolutional codes from block codes and use it
to derive families of non-catastrophic quantum convolutional codes from BCH
codes.
The dissertation concludes with a discussion of some open problems
Constructions of Quantum Convolutional Codes
We address the problems of constructing quantum convolutional codes (QCCs)
and of encoding them. The first construction is a CSS-type construction which
allows us to find QCCs of rate 2/4. The second construction yields a quantum
convolutional code by applying a product code construction to an arbitrary
classical convolutional code and an arbitrary quantum block code. We show that
the resulting codes have highly structured and efficient encoders. Furthermore,
we show that the resulting quantum circuits have finite depth, independent of
the lengths of the input stream, and show that this depth is polynomial in the
degree and frame size of the code.Comment: 5 pages, to appear in the Proceedings of the 2007 IEEE International
Symposium on Information Theor
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