119,816 research outputs found
Simple Quantum Error Correcting Codes
Methods of finding good quantum error correcting codes are discussed, and
many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where
C_1 and C_2 are classical codes, is used to obtain codes for up to 16
information qubits with correction of small numbers of errors. The results are
tabulated. More efficient codes are obtained by allowing C_1 to have reduced
distance, and introducing sign changes among the code words in a systematic
manner. This systematic approach leads to single-error correcting codes for 3,
4 and 5 information qubits with block lengths of 8, 10 and 11 qubits
respectively.Comment: Submitted to Phys. Rev. A. in May 1996. 21 pages, no figures. Further
information at http://eve.physics.ox.ac.uk/ASGhome.htm
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
Codes for the Quantum Erasure Channel
The quantum erasure channel (QEC) is considered. Codes for the QEC have to
correct for erasures, i. e., arbitrary errors at known positions. We show that
four qubits are necessary and sufficient to encode one qubit and correct one
erasure, in contrast to five qubits for unknown positions. Moreover, a family
of quantum codes for the QEC, the quantum BCH codes, that can be efficiently
decoded is introduced.Comment: 6 pages, RevTeX, no figures, submitted to Physical Review A, code
extended to encode 2 qubits, references adde
Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction
A method for concatenating quantum error-correcting codes is presented. The
method is applicable to a wide class of quantum error-correcting codes known as
Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate
in the Shannon theoretic sense and that are decodable in polynomial time are
presented. The rate is the highest among those known to be achievable by CSS
codes. Moreover, the best known lower bound on the greatest minimum distance of
codes constructible in polynomial time is improved for a wide range.Comment: 16 pages, 3 figures. Ver.4: Title changed. Ver.3: Due to a request of
the AE of the journal, the present version has become a combination of
(thoroughly revised) quant-ph/0610194 and the former quant-ph/0610195.
Problem formulations of polynomial complexity are strictly followed. An
erroneous instance of a lower bound on minimum distance was remove
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