Bounds on Quantum Error Correcting Codes

2008 ◽  
pp. 223-260
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error

[[2n,n]] Additive quantum error correcting codes

2008 ◽  
Vol 25 (4) ◽  
pp. 519-522
Author(s):  
Yongjun Du ◽  
Yuefei Ma
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error

scholarly journals Efficiently computing logical noise in quantum error-correcting codes

2021 ◽  
Vol 103 (6) ◽  
Author(s):  
Stefanie J. Beale ◽  
Joel J. Wallman
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error

ON TWO PROBLEMS OF ASYMMETRIC QUANTUM CODES

2014 ◽  
Vol 28 (06) ◽  
pp. 1450017 ◽  
Author(s):  
RUIHU LI ◽  
GEN XU ◽  
LUOBIN GUO
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Bch Code ◽  
Quantum Codes ◽  
Quantum Code ◽  
Asymmetric Quantum ◽  
Asymmetric Quantum Codes ◽  
Special Cases ◽  
Quantum Error ◽  
Better Than

In this paper, we discuss two problems on asymmetric quantum error-correcting codes (AQECCs). The first one is on the construction of a [[12, 1, 5/3]]2 asymmetric quantum code, we show an impure [[12, 1, 5/3 ]]2 exists. The second one is on the construction of AQECCs from binary cyclic codes, we construct many families of new asymmetric quantum codes with dz> δ max +1 from binary primitive cyclic codes of length n = 2m-1, where δ max = 2⌈m/2⌉-1 is the maximal designed distance of dual containing narrow sense BCH code of length n = 2m-1. A number of known codes are special cases of the codes given here. Some of these AQECCs have parameters better than the ones available in the literature.

scholarly journals Transforming graph states using single-qubit operations

2018 ◽  
Vol 376 (2123) ◽  
pp. 20170325 ◽  
Author(s):  
Axel Dahlberg ◽  
Stephanie Wehner
Keyword(s):  
Quantum Information ◽  
Error Correcting Codes ◽  
Graph States ◽  
Target State ◽  
Single Qubit ◽  
Minor Problem ◽  
Fixed Parameter ◽  
Source State ◽  
Quantum Error ◽  
Stabilizer States

Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

scholarly journals Theory of quantum error-correcting codes

1997 ◽  
Vol 55 (2) ◽  
pp. 900-911 ◽  
Author(s):  
Emanuel Knill ◽  
Raymond Laflamme
Keyword(s):  
Error Correcting Codes ◽  
Quantum Error

scholarly journals Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

2015 ◽  
Vol 2015 (6) ◽  
Author(s):  
Fernando Pastawski ◽  
Beni Yoshida ◽  
Daniel Harlow ◽  
John Preskill
Keyword(s):  
Error Correcting Codes ◽  
Boundary Correspondence ◽  
Quantum Error

Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1107-1116
Author(s):  
Ruihu Li ◽  
Luobin Guo ◽  
Zongben Xu
Keyword(s):  
Error Correction ◽  
Infinite Family ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Singleton Bound ◽  
Minimum Distances ◽  
Quantum Error

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices

2018 ◽  
Vol 18 (3&4) ◽  
pp. 223-230
Author(s):  
Xianmang He
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Minimum Distance ◽  
Quantum Information Theory ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Error ◽  
Generator Matrices ◽  
Quantum Mds Codes

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

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