2,195 research outputs found
Quantum Convolutional Error Correction Codes
I report two general methods to construct quantum convolutional codes for
quantum registers with internal states. Using one of these methods, I
construct a quantum convolutional code of rate 1/4 which is able to correct one
general quantum error for every eight consecutive quantum registers.Comment: To be reported in the 1st NASA Conf. on Quantum Comp., uses
llncs.sty, 12 page
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
Description of a quantum convolutional code
We describe a quantum error correction scheme aimed at protecting a flow of
quantum information over long distance communication. It is largely inspired by
the theory of classical convolutional codes which are used in similar
circumstances in classical communication. The particular example shown here
uses the stabilizer formalism, which provides an explicit encoding circuit. An
associated error estimation algorithm is given explicitly and shown to provide
the most likely error over any memoryless quantum channel, while its complexity
grows only linearly with the number of encoded qubits.Comment: 4 pages, uses revtex4. Minor correction in the encoding and decoding
circuit
Quantum Coding with Entanglement
Quantum error-correcting codes will be the ultimate enabler of a future
quantum computing or quantum communication device. This theory forms the
cornerstone of practical quantum information theory. We provide several
contributions to the theory of quantum error correction--mainly to the theory
of "entanglement-assisted" quantum error correction where the sender and
receiver share entanglement in the form of entangled bits (ebits) before
quantum communication begins. Our first contribution is an algorithm for
encoding and decoding an entanglement-assisted quantum block code. We then give
several formulas that determine the optimal number of ebits for an
entanglement-assisted code. The major contribution of this thesis is the
development of the theory of entanglement-assisted quantum convolutional
coding. A convolutional code is one that has memory and acts on an incoming
stream of qubits. We explicitly show how to encode and decode a stream of
information qubits with the help of ancilla qubits and ebits. Our
entanglement-assisted convolutional codes include those with a
Calderbank-Shor-Steane structure and those with a more general structure. We
then formulate convolutional protocols that correct errors in noisy
entanglement. Our final contribution is a unification of the theory of quantum
error correction--these unified convolutional codes exploit all of the known
resources for quantum redundancy.Comment: Ph.D. Thesis, University of Southern California, 2008, 193 pages, 2
tables, 12 figures, 9 limericks; Available at
http://digitallibrary.usc.edu/search/controller/view/usctheses-m1491.htm
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
Good Quantum Convolutional Error Correction Codes And Their Decoding Algorithm Exist
Quantum convolutional code was introduced recently as an alternative way to
protect vital quantum information. To complete the analysis of quantum
convolutional code, I report a way to decode certain quantum convolutional
codes based on the classical Viterbi decoding algorithm. This decoding
algorithm is optimal for a memoryless channel. I also report three simple
criteria to test if decoding errors in a quantum convolutional code will
terminate after a finite number of decoding steps whenever the Hilbert space
dimension of each quantum register is a prime power. Finally, I show that
certain quantum convolutional codes are in fact stabilizer codes. And hence,
these quantum stabilizer convolutional codes have fault-tolerant
implementations.Comment: Minor changes, to appear in PR
The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing
and protecting fragile qubits against the undesirable effects of quantum
decoherence. Similar to classical codes, hashing bound approaching QECCs may be
designed by exploiting a concatenated code structure, which invokes iterative
decoding. Therefore, in this paper we provide an extensive step-by-step
tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided
concatenated quantum codes based on the underlying quantum-to-classical
isomorphism. These design lessons are then exemplified in the context of our
proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the
outer component of a concatenated quantum code. The proposed QIRCC can be
dynamically adapted to match any given inner code using EXIT charts, hence
achieving a performance close to the hashing bound. It is demonstrated that our
QIRCC-based optimized design is capable of operating within 0.4 dB of the noise
limit
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