A novel construction for quantum stabilizer codes based on binary formalism

2020 ◽  
Vol 34 (08) ◽  
pp. 2050059 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Minimum Distance ◽  
Binary Vector ◽  
Quantum Codes ◽  
Quantum Code ◽  
Parity Check Matrix ◽  
Stabilizer Codes ◽  
Check Matrix ◽  
Stabilizer Code ◽  
Quantum Stabilizer Code ◽  
Quantum Stabilizer Codes

In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.

scholarly journals New Constructions of Quantum Stabilizer Codes Based on Difference Sets

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 655 ◽  
Author(s):  
Duc Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Difference Sets ◽  
Combinatorial Design ◽  
Inner Product ◽  
Code Construction ◽  
Stabilizer Codes ◽  
Stabilizer Code ◽  
The Difference ◽  
Symplectic Inner Product ◽  
Quantum Stabilizer Code ◽  
Quantum Stabilizer Codes

In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is first explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.

scholarly journals Weight reduction for quantum codes

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1307-1334
Author(s):  
Mathew B. Hastings
Keyword(s):  
Weight Reduction ◽  
High Dimensional ◽  
Quantum Codes ◽  
Stabilizer Codes ◽  
Linear Distance ◽  
Logical Qubits ◽  
Stabilizer Code ◽  
Logarithmic Weight

We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights O(1) and such that O(1) generators act on any given qubit. The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and “cosoundness” (defined later). Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers. Assuming a conjecture in geometry[11], this allows the construction of CSS stabilizer codes with generator weight O(1) and almost linear distance. Another application of the construction is to increasing the distance to X or Z errors, whichever is smaller, so that the two distances are equal.

Maximal entanglement entanglement-assisted quantum codes of distance three

2015 ◽  
Vol 13 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Luobin Guo ◽  
Qiang Fu ◽  
Ruihu Li ◽  
Liangdong Lu
Keyword(s):  
Minimum Distance ◽  
Error Correcting Code ◽  
Quantum Codes ◽  
Quantum Code ◽  
Maximal Entanglement ◽  
Dual Distance ◽  
Recursive Constructions ◽  
Almost All ◽  
Quantum Error

Entanglement-assisted quantum error correcting code (EAQECC) is a generalization of standard stabilizer quantum code. Maximal entanglement EAQECCs can achieve the EA-hashing bound asymptotically. In this work, we give elementary recursive constructions of quaternary zero radical codes with dual distance three for all n ≥ 4. Consequently, good maximal entanglement EAQECCs of minimum distance three for such length n are obtained. Almost all of these EAQECCs are optimal or near optimal according to the EA-quantum Hamming bound.

scholarly journals Quantum stabilizer codes from Abelian and non-Abelian groups association schemes

2015 ◽  
Vol 13 (03) ◽  
pp. 1550021 ◽  
Author(s):  
Avaz Naghipour ◽  
Mohammad Ali Jafarizadeh ◽  
Sedaghat Shahmorad
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Regular Representation ◽  
Association Schemes ◽  
Stabilizer Codes ◽  
Cyclic Groups ◽  
Minimum Distances ◽  
Quantum Stabilizer Code ◽  
Group Association ◽  
Quantum Stabilizer Codes

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

scholarly journals SIMPLE FAULT-TOLERANT ENCODING OVER q-ARY CSS QUANTUM CODES

2007 ◽  
Vol 05 (05) ◽  
pp. 705-716
Author(s):  
PEDRO J. SALAS
Keyword(s):  
Fault Tolerant ◽  
Quantum Codes ◽  
Quantum Code ◽  
Simple Method ◽  
Stabilizer Codes ◽  
Css Codes ◽  
Quantum Computations

CSS codes are a subfamily of stabilizer codes especially appropriate for fault-tolerant quantum computations. A very simple method is proposed to encode a general qudit when a Calderbank–Shor–Steane quantum code, defined over a q-ary alphabet, is used.

COMBINATORIAL CONSTRUCTION OF LINEAR QUANTUM CODES FROM QUATERNARY BCH CODES

2009 ◽  
Vol 07 (05) ◽  
pp. 1039-1046 ◽  
Author(s):  
RUIHU LI ◽  
ZONGBEN XU
Keyword(s):  
Bch Codes ◽  
Quantum Codes ◽  
Dual Codes ◽  
Stabilizer Codes ◽  
Combinatorial Construction ◽  
Quantum Stabilizer Codes ◽  
Linear Quantum

Classical BCH codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes. But for given restricted length n, good quantum BCH codes are very sparse. In this paper, by puncturing and pasting check matrices of Hermitian dual containing BCH codes over the quaternary field, we construct many linear quantum codes with good parameters, and some of them have parameters exceeding the finite Gilbert-Varshamov bound for stabilizer quantum codes.

Quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over GF(4)

2020 ◽  
Vol 34 (05) ◽  
pp. 2050017 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Linear Codes ◽  
Galois Field ◽  
Inner Product ◽  
Quantum Codes ◽  
Product Codes ◽  
Stabilizer Codes ◽  
Comparison Results ◽  
Construction Of Self ◽  
New Construction ◽  
Quantum Stabilizer Codes

In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum 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.

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