697 research outputs found
How Much Entanglement Does a Quantum Code Need?
In the setting of entanglement-assisted quantum error-correcting codes
(EAQECCs), the sender and the receiver have access to pre-shared entanglement.
Such codes promise better information rates or improved error handling
properties. Entanglement incurs costs and must be judiciously calibrated in
designing quantum codes with good performance, relative to their deployment
parameters.
Revisiting known constructions, we devise tools from classical coding theory
to better understand how the amount of entanglement can be varied. We present
three new propagation rules and discuss how each of them affects the error
handling. Tables listing the parameters of the best performing qubit and qutrit
EAQECCs that we can explicitly construct are supplied for reference and
comparison
Entanglement-Assisted Quantum Quasi-Cyclic Low-Density Parity-Check Codes
We investigate the construction of quantum low-density parity-check (LDPC)
codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or
equal to 6. We have shown that the classical codes in the generalized
Calderbank-Shor-Steane (CSS) construction do not need to satisfy the
dual-containing property as long as pre-shared entanglement is available to
both sender and receiver. We can use this to avoid the many 4-cycles which
typically arise in dual-containing LDPC codes. The advantage of such quantum
codes comes from the use of efficient decoding algorithms such as sum-product
algorithm (SPA). It is well known that in the SPA, cycles of length 4 make
successive decoding iterations highly correlated and hence limit the decoding
performance. We show the principle of constructing quantum QC-LDPC codes which
require only small amounts of initial shared entanglement.Comment: 8 pages, 1 figure. Final version that will show up on PRA. Minor
changes in contents and Titl
Characterization and mass formulas of symplectic self-orthogonal and LCD codes and their application
The object of this paper is to study two very important classes of codes in
coding theory, namely self-orthogonal (SO) and linear complementary dual (LCD)
codes under the symplectic inner product, involving characterization,
constructions, and their application. Using such a characterization, we
determine the mass formulas of symplectic SO and LCD codes by considering the
action of the symplectic group, and further obtain some asymptotic results.
Finally, under the Hamming distance, we obtain some symplectic SO (resp. LCD)
codes with improved parameters directly compared with Euclidean SO (resp. LCD)
codes. Under the symplectic distance, we obtain some additive SO (resp.
additive complementary dual) codes with improved parameters directly compared
with Hermitian SO (resp. LCD) codes. Further, we also construct many good
additive codes outperform the best-known linear codes in Grassl's code table.
As an application, we construct a number of record-breaking
(entanglement-assisted) quantum error-correcting codes, which improve Grassl's
code table
Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted
quantum low-density parity-check (LDPC) codes, which is based on combinatorial
design theory. Explicit constructions are given for entanglement-assisted
quantum error-correcting codes (EAQECCs) with many desirable properties. These
properties include the requirement of only one initial entanglement bit, high
error correction performance, high rates, and low decoding complexity. The
proposed method produces infinitely many new codes with a wide variety of
parameters and entanglement requirements. Our framework encompasses various
codes including the previously known entanglement-assisted quantum LDPC codes
having the best error correction performance and many new codes with better
block error rates in simulations over the depolarizing channel. We also
determine important parameters of several well-known classes of quantum and
classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review
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
Duality in Entanglement-Assisted Quantum Error Correction
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is
defined from the orthogonal group of a simplified stabilizer group. From the
Poisson summation formula, this duality leads to the MacWilliams identities and
linear programming bounds for EAQEC codes. We establish a table of upper and
lower bounds on the minimum distance of any maximal-entanglement EAQEC code
with length up to 15 channel qubits.Comment: This paper is a compact version of arXiv:1010.550
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