scholarly journals GRAPHICAL QUANTUM ERROR-CORRECTING CODES BASED ON ENTANGLEMENT OF SUBGRAPHS

2012 ◽  
Vol 26 (04) ◽  
pp. 1250024
Author(s):  
YUAN LI ◽  
MANTAO XU ◽  
QIANG SUN
Keyword(s):  
Error Correction ◽  
Error Correcting Codes ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Quantum Code ◽  
Generator Matrix ◽  
Code Construction ◽  
Nested Codes ◽  
Colorable Graph ◽  
Quantum Error

Graphical approach provides a more intuitive and simple way to construct error correction codes. How to obtain generator matrix is the key problem of constructing graphical quantum codes. In this paper, we further generalize the graphical quantum code construction method by entangling its disconnected subgraphs, so that the corresponding generator matrix of quantum nondegenerate codes can be easily obtained. By making use of the method of subgraphs entangling, we also point out its application in adjacency matrix constructions of larger colorable graph and graphical quantum nested codes.

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 Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes

2021 ◽  
Author(s):  
Dongsheng Wang ◽  
Yunjiang Wang ◽  
Ningping Cao ◽  
Bei Zeng ◽  
Raymond Lafflamme
Keyword(s):  
Error Correction ◽  
Quantum Computation ◽  
Fault Tolerant ◽  
Valence Bond ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Exact Quantum ◽  
Valence Bond Solid ◽  
Quantum Error

Abstract In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes (``quasi codes''). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.

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.

scholarly journals A combinatorial approach to quantum error correcting codes

2014 ◽  
Vol 06 (04) ◽  
pp. 1450054
Author(s):  
German Luna ◽  
Samuel Reid ◽  
Bianca De Sanctis ◽  
Vlad Gheorghiu
Keyword(s):  
Error Correcting Code ◽  
Combinatorial Problem ◽  
Error Correcting Codes ◽  
Quantum Code ◽  
Combinatorial Approach ◽  
Graph Colorings ◽  
Explicit Algorithm ◽  
Colorable Graph ◽  
Quantum Error

Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric n-vertices colorable graph and a group of operations (coloring rules) on the graph: find the minimum sequence of operations that maps between two given graph colorings. We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code. Computing the distance of a quantum code is a highly non-trivial problem and our method may be of use in the construction of better codes.

Quantum error correction nested codes derived from jacket transform

2019 ◽  
Vol 33 (31) ◽  
pp. 1950378
Author(s):  
Yuan Li ◽  
Niansheng Chen ◽  
Yiyuan Luo
Keyword(s):  
Error Correction ◽  
Decomposition Methods ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Arbitrary Size ◽  
Novel Approach ◽  
Nested Codes ◽  
Quantum Error

A novel approach for construction of quantum codes employing jacket transform is proposed in this paper. The constructed quantum nested codes has less complexities than the previous quantum codes in encoding procedures as well as decoding procedures. Furthermore, the quantum error codes can be constructed with an arbitrary size using the number decomposition, and the corresponding distance of constructed quantum codes can be adjusted by changing decomposition methods.

scholarly journals From Quantum Codes to Gravity: A Journey of Gravitizing Quantum Mechanics

Universe ◽  
2021 ◽  
Vol 8 (1) ◽  
pp. 1
Author(s):  
Chun-Jun Cao
Keyword(s):  
Quantum Mechanics ◽  
Quantum Gravity ◽  
Error Correction ◽  
Quantum Error Correction ◽  
Quantum States ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Recent Approach ◽  
Tensor Network ◽  
Quantum Error

In this note, I review a recent approach to quantum gravity that “gravitizes” quantum mechanics by emerging geometry and gravity from complex quantum states. Drawing further insights from tensor network toy models in AdS/CFT, I propose that approximate quantum error correction codes, when re-adapted into the aforementioned framework, also have promise in emerging gravity in near-flat geometries.

scholarly journals Fast Constructions of Quantum Codes Based on Residues Pauli Block Matrices

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Ying Guo ◽  
Guihu Zeng ◽  
MoonHo Lee
Keyword(s):  
Error Correction ◽  
Abelian Groups ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Block Matrices ◽  
Quadratic Residues ◽  
Quantum Error Correction Codes ◽  
Block Transforms ◽  
Quantum Error

We demonstrate how to fast construct quantum error-correction codes based on quadratic residues Pauli block transforms. The present quantum codes have an advantage of being fast designed from Abelian groups on the basis of Pauli block matrices that can be yielded from quadratic residues with much efficiency.

On the iterative decoding of sparse quantum codes

2008 ◽  
Vol 8 (10) ◽  
pp. 986-1000
Author(s):  
D. Poulin ◽  
Y. Chung
Keyword(s):  
Iterative Decoding ◽  
Belief Propagation ◽  
Quantum Codes ◽  
Error Correction Codes ◽  
Heuristic Methods ◽  
Tanner Graph ◽  
Coding Scheme ◽  
Propagation Algorithm ◽  
Belief Propagation Algorithm ◽  
Quantum Error

We address the problem of decoding sparse quantum error correction codes. For Pauli channels, this task can be accomplished by a version of the belief propagation algorithm used for decoding sparse classical codes. Quantum codes pose two new challenges however. Firstly, their Tanner graph unavoidably contain small loops which typically undermines the performance of belief propagation. Secondly, sparse quantum codes are by definition highly degenerate. The standard belief propagation algorithm does not exploit this feature, but rather it is impaired by it. We propose heuristic methods to improve belief propagation decoding, specifically targeted at these two problems. While our results exhibit a clear improvement due to the proposed heuristic methods, they also indicate that the main source of errors in the quantum coding scheme remains in the decoding.

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