scholarly journals Reinterpreting the Stabilizer Formalism

2021 ◽  
Author(s):  
Vatsal Pramod Jha ◽  
Udaya Parampalli ◽  
Abhay Kumar Singh
Keyword(s):  
Binary Codes ◽  
Quantum Codes ◽  
Stabilizer Codes ◽  
Orthogonal Codes ◽  
Css Construction

<div>Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.</div>

scholarly journals Reinterpreting the Stabilizer Formalism

2021 ◽  
Author(s):  
Vatsal Pramod Jha ◽  
Udaya Parampalli ◽  
Abhay Kumar Singh
Keyword(s):  
Binary Codes ◽  
Quantum Codes ◽  
Stabilizer Codes ◽  
Orthogonal Codes ◽  
Css Construction

<div>Stabilizer codes, introduced in [2], [3], have been a prominent example of quantum codes constructed via classical codes. The paper [3], introduces the stabilizer formalism for obtaining additive quantum codes of length n from Hermitian self-orthogonal codes of length n over GF(4). In the present work, we reinterpret the stabilizer formalism by considering binary codes over the symbol-pair metric (see [9]). Specifically, the present work constructs additive quantum codes of length n from certain binary codes of length n considered over the symbol-pair metric. We also present the Modified CSS Construction which is used to obtain quantum codes with parameters.</div>

scholarly journals Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1495
Author(s):  
Yansheng Wu ◽  
Yoonjin Lee
Keyword(s):  
Linear Codes ◽  
Sufficient Conditions ◽  
Quantum Codes ◽  
Binary Linear Codes ◽  
Hamming Codes ◽  
Orthogonal Codes ◽  
Css Codes ◽  
Css Construction ◽  
Necessary And Sufficient ◽  
Quantum Error

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_4 + u\mathbb{F}_4$

2011 ◽  
Vol 09 (02) ◽  
pp. 689-700 ◽  
Author(s):  
XIAOSHAN KAI ◽  
SHIXIN ZHU
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Binary Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Orthogonal Codes

We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.

scholarly journals New quantum codes from skew constacyclic codes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ram Krishna Verma ◽  
Om Prakash ◽  
Ashutosh Singh ◽  
Habibul Islam
Keyword(s):  
Finite Field ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Chain Ring ◽  
Css Construction ◽  
Positive Integers

<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id="M7">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id="M8">\begin{document}$ p^{2^{\ell} m} $\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id="M10">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>

scholarly journals Quantum Codes Constructed from Self-Dual Codes and Maximal Self-Orthogonal Codes Over F5

2012 ◽  
Vol 29 ◽  
pp. 3448-3453
Author(s):  
Luobin Guo ◽  
Yuena Ma ◽  
Youqian Feng
Keyword(s):  
Quantum Codes ◽  
Dual Codes ◽  
Orthogonal Codes

scholarly journals C_3, semi-Clifford and genralized semi-Clifford operations

2010 ◽  
Vol 10 (1&2) ◽  
pp. 41-59
Author(s):  
S. Beigi ◽  
P.W. Shor
Keyword(s):  
Quantum Computation ◽  
Basic Problem ◽  
Fault Tolerant ◽  
Quantum Codes ◽  
Stabilizer Codes

Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called ${\mathcal{C}}_k$ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). ${\mathcal{C}}_k$ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of Clifford gates. Zeng et al. in (Phys. Rev. A 77, 042313) raise the question of the relation between ${\mathcal{C}}_k$ hierarchy and the set of semi-Clifford and generalized semi-Clifford operators. They conjecture that any ${\mathcal{C}}_k$ gate is a generalized semi-Clifford operator. In this paper, we prove this conjecture for $k=3$. Using the techniques that we develop, we obtain more insight on how to characterize ${\mathcal{C}}_3$ gates. Indeed, the more we understand ${\mathcal{C}}_3$, the more intuition we have on ${\mathcal{C}}_k$, $k\geq 4$, and then we have a way of attacking the conjecture for larger $k$.

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.

scholarly journals Flag fault-tolerant error correction with arbitrary distance codes

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 53 ◽  
Author(s):  
Christopher Chamberland ◽  
Michael E. Beverland
Keyword(s):  
Error Correction ◽  
Fault Tolerant ◽  
Quantum Error Correction ◽  
Quantum Codes ◽  
Parity Check ◽  
Code Length ◽  
Stabilizer Codes ◽  
Low Density Parity Check ◽  
Quantum Error ◽  
First Time

In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting fromvfaults in the circuit have weight greater thanv. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.

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.

Sign in / Sign up

Export Citation Format

Share Document