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Enlargement of Calderbank-Shor-Steane quantum codes
, 1999
"... It is shown that a classical error correcting code C = [n,k,d] which contains its dual, C ⊥ ⊆ C, and which can be enlarged to C ′ = [n,k ′> k + 1,d ′], can be converted into a quantum code of parameters [[n,k + k ′ − n,min(d, ⌈3d ′ /2⌉)]]. This is a generalisation of a previous construction, it ..."
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It is shown that a classical error correcting code C = [n,k,d] which contains its dual, C ⊥ ⊆ C, and which can be enlarged to C ′ = [n,k ′> k + 1,d ′], can be converted into a quantum code of parameters [[n,k + k ′ − n,min(d, ⌈3d ′ /2⌉)]]. This is a generalisation of a previous construction, it enables many new codes of good efficiency to be discovered. Examples based on classical Bose Chaudhuri Hocquenghem (BCH) codes are discussed.
Introduction to quantum error correction
, 1998
"... An introduction to quantum error correction (QEC) is given, and some recent developments ..."
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Cited by 15 (0 self)
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An introduction to quantum error correction (QEC) is given, and some recent developments
Nonbinary quantum Reed-Muller codes
- In Proc. Int. Symp. Inform. Theory
, 2005
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On Quantum and Classical Error Control Codes: Constructions and Applications
, 812
"... degree of doctoral of Philosophy on Fall 2007. Some other parts were added later without peer reviews. The document is reformatted. Please report all typos or errors to the author with all due haste. It is conjectured that quantum computers are able to solve certain problems more quickly than any de ..."
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degree of doctoral of Philosophy on Fall 2007. Some other parts were added later without peer reviews. The document is reformatted. Please report all typos or errors to the author with all due haste. It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. For instance, Shor’s algorithm is able to factor large integers in polynomial time on a quantum computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. Therefore, one has to mitigate the resulting noise and decoherence effects to avoid computational errors. In this work, I study various aspects of quantum error control codes – the key component of fault-tolerant quantum information processing. I present the fundamental theory and necessary background of quantum codes and construct many families of quantum block and convolutional codes over finite fields, in addition to families of subsystem codes. This work is organized into these parts: Quantum Block Codes. After introducing the theory of quantum block codes, I establish conditions when BCH codes are self-orthogonal (or dual-containing) with respect to Euclidean and Hermitian inner
Nonbinary Stabilizer Codes
"... Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposi ..."
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Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
2.1 Approximate Threshold Scaling................................ 4
, 2009
"... We study a comprehensive list of quantum codes as candidates of codes to be used at the bottom, physical, level in a fault-tolerant code architecture. Using the Aliferis-Gottesman-Preskill (AGP) ex-Rec method we calculate the pseudo-threshold for these codes against depolarizing noise at various lev ..."
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We study a comprehensive list of quantum codes as candidates of codes to be used at the bottom, physical, level in a fault-tolerant code architecture. Using the Aliferis-Gottesman-Preskill (AGP) ex-Rec method we calculate the pseudo-threshold for these codes against depolarizing noise at various levels of overhead. We estimate the logical noise rate as a function of overhead at a physical error rate of
