124 research outputs found
Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error-Correcting Codes
Simple rate-1/3 single-error-correcting unrestricted and CSS-type quantum
convolutional codes are constructed from classical self-orthogonal
\F_4-linear and \F_2-linear convolutional codes, respectively. These
quantum convolutional codes have higher rate than comparable quantum block
codes or previous quantum convolutional codes, and are simple to decode. A
block single-error-correcting [9, 3, 3] tail-biting code is derived from the
unrestricted convolutional code, and similarly a [15, 5, 3] CSS-type block code
from the CSS-type convolutional code.Comment: 5 pages; to appear in Proceedings of 2005 IEEE International
Symposium on Information Theor
Convolutional and tail-biting quantum error-correcting codes
Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to
4096 states and minimum distances up to 10 are constructed as stabilizer codes
from classical self-orthogonal rate-1/n F_4-linear and binary linear
convolutional codes, respectively. These codes generally have higher rate and
less decoding complexity than comparable quantum block codes or previous
quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same
rate and error-correction capability and essentially the same decoding
algorithms are derived from these convolutional codes via tail-biting.Comment: 30 pages. Submitted to IEEE Transactions on Information Theory. Minor
revisions after first round of review
Quantum Block and Convolutional Codes from Self-orthogonal Product Codes
We present a construction of self-orthogonal codes using product codes. From
the resulting codes, one can construct both block quantum error-correcting
codes and quantum convolutional codes. We show that from the examples of
convolutional codes found, we can derive ordinary quantum error-correcting
codes using tail-biting with parameters [[42N,24N,3]]_2. While it is known that
the product construction cannot improve the rate in the classical case, we show
that this can happen for quantum codes: we show that a code [[15,7,3]]_2 is
obtained by the product of a code [[5,1,3]]_2 with a suitable code.Comment: 5 pages, paper presented at the 2005 IEEE International Symposium on
Information Theor
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
Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes
We present an algorithm to construct quantum circuits for encoding and
inverse encoding of quantum convolutional codes. We show that any quantum
convolutional code contains a subcode of finite index which has a
non-catastrophic encoding circuit. Our work generalizes the conditions for
non-catastrophic encoders derived in a paper by Ollivier and Tillich
(quant-ph/0401134) which are applicable only for a restricted class of quantum
convolutional codes. We also show that the encoders and their inverses
constructed by our method naturally can be applied online, i.e., qubits can be
sent and received with constant delay.Comment: 6 pages, 1 figure, submitted to 2006 IEEE International Symposium on
Information Theor
Examples of minimal-memory, non-catastrophic quantum convolutional encoders
One of the most important open questions in the theory of quantum
convolutional coding is to determine a minimal-memory, non-catastrophic,
polynomial-depth convolutional encoder for an arbitrary quantum convolutional
code. Here, we present a technique that finds quantum convolutional encoders
with such desirable properties for several example quantum convolutional codes
(an exposition of our technique in full generality will appear elsewhere). We
first show how to encode the well-studied Forney-Grassl-Guha (FGG) code with an
encoder that exploits just one memory qubit (the former Grassl-Roetteler
encoder requires 15 memory qubits). We then show how our technique can find an
online decoder corresponding to this encoder, and we also detail the operation
of our technique on a different example of a quantum convolutional code.
Finally, the reduction in memory for the FGG encoder makes it feasible to
simulate the performance of a quantum turbo code employing it, and we present
the results of such simulations.Comment: 5 pages, 2 figures, Accepted for the International Symposium on
Information Theory 2011 (ISIT 2011), St. Petersburg, Russia; v2 has minor
change
Minimal-memory realization of pearl-necklace encoders of general quantum convolutional codes
Quantum convolutional codes, like their classical counterparts, promise to
offer higher error correction performance than block codes of equivalent
encoding complexity, and are expected to find important applications in
reliable quantum communication where a continuous stream of qubits is
transmitted. Grassl and Roetteler devised an algorithm to encode a quantum
convolutional code with a "pearl-necklace encoder." Despite their theoretical
significance as a neat way of representing quantum convolutional codes, they
are not well-suited to practical realization. In fact, there is no
straightforward way to implement any given pearl-necklace structure. This paper
closes the gap between theoretical representation and practical implementation.
In our previous work, we presented an efficient algorithm for finding a
minimal-memory realization of a pearl-necklace encoder for
Calderbank-Shor-Steane (CSS) convolutional codes. This work extends our
previous work and presents an algorithm for turning a pearl-necklace encoder
for a general (non-CSS) quantum convolutional code into a realizable quantum
convolutional encoder. We show that a minimal-memory realization depends on the
commutativity relations between the gate strings in the pearl-necklace encoder.
We find a realization by means of a weighted graph which details the
non-commutative paths through the pearl-necklace. The weight of the longest
path in this graph is equal to the minimal amount of memory needed to implement
the encoder. The algorithm has a polynomial-time complexity in the number of
gate strings in the pearl-necklace encoder.Comment: 16 pages, 5 figures; extends paper arXiv:1004.5179v
Quantum convolutional data-syndrome codes
We consider performance of a simple quantum convolutional code in a
fault-tolerant regime using several syndrome measurement/decoding strategies
and three different error models, including the circuit model.Comment: Abstract submitted for The 20th IEEE International Workshop on Signal
Processing Advances in Wireless Communications (SPAWC 2019
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