872 research outputs found
Catalytic quantum error correction
We develop the theory of entanglement-assisted quantum error correcting
(EAQEC) codes, a generalization of the stabilizer formalism to the setting in
which the sender and receiver have access to pre-shared entanglement.
Conventional stabilizer codes are equivalent to dual-containing symplectic
codes. In contrast, EAQEC codes do not require the dual-containing condition,
which greatly simplifies their construction. We show how any quaternary
classical code can be made into a EAQEC code. In particular, efficient modern
codes, like LDPC codes, which attain the Shannon capacity, can be made into
EAQEC codes attaining the hashing bound. In a quantum computation setting,
EAQEC codes give rise to catalytic quantum codes which maintain a region of
inherited noiseless qubits.
We also give an alternative construction of EAQEC codes by making classical
entanglement assisted codes coherent.Comment: 30 pages, 10 figures. Notation change: [[n,k;c]] instead of
[[n,k-c;c]
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
Entanglement-assisted Coding Theory
In this dissertation, I present a general method for studying quantum error
correction codes (QECCs). This method not only provides us an intuitive way of
understanding QECCs, but also leads to several extensions of standard QECCs,
including the operator quantum error correction (OQECC), the
entanglement-assisted quantum error correction (EAQECC). Furthermore, we can
combine both OQECC and EAQECC into a unified formalism, the
entanglement-assisted operator formalism. This provides great flexibility of
designing QECCs for different applications. Finally, I show that the
performance of quantum low-density parity-check codes will be largely improved
using entanglement-assisted formalism.Comment: PhD dissertation, 102 page
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
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
Optimal quaternary linear codes with one-dimensional Hermitian hull and the related EAQECCs
Linear codes with small hulls over finite fields have been extensively
studied due to their practical applications in computational complexity and
information protection. In this paper, we develop a general method to determine
the exact value of for or , where denotes the largest minimum
distance among all quaternary linear codes with one-dimensional
Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and
Jitman on the largest minimum distance of a quaternary linear code with
one-dimensional Hermitian hull. As an application, we construct some binary
entanglement-assisted quantum error-correcting codes (EAQECCs) from quaternary
linear codes with one-dimensional Hermitian hull. Some of these EAQECCs are
optimal codes, and some of them are better than previously known ones.Comment: arXiv admin note: text overlap with arXiv:2211.0248
The hull of two classical propagation rules and their applications
Propagation rules are of great help in constructing good linear codes. Both
Euclidean and Hermitian hulls of linear codes perform an important part in
coding theory. In this paper, we consider these two aspects together and
determine the dimensions of Euclidean and Hermitian hulls of two classical
propagation rules, namely, the direct sum construction and the
-construction. Some new criteria for resulting codes
derived from these two propagation rules being self-dual, self-orthogonal or
linear complement dual (LCD) codes are given. As applications, we construct
some linear codes with prescribed hull dimensions and many new binary, ternary
Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD
codes and good quaternary Hermitian LCD codes which are optimal or have best or
almost best known parameters according to Datebase at
. Moreover, our methods contributes positively to
improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table
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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