A Novel and Efficient Stabilizer Codes Over NonCyclic Hadamard Difference Sets for Quantum System.

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Bibliographic Details
Title:	A Novel and Efficient Stabilizer Codes Over NonCyclic Hadamard Difference Sets for Quantum System.
Authors:	Goswami, Shivender¹ shivendrgoswami@gmail.com, Kumar, Manoj² sdmkg1@gmail.com, Mishra, R. K.³ rkmsit@rediffmail.com, Rathor, Akash¹ akashrathor9760@gmail.com
Source:	IAENG International Journal of Applied Mathematics. Jul2024, Vol. 54 Issue 7, p1416-1426. 11p.
Subjects:	Difference sets, Parity-check matrix, Binary operations, Circulant matrices, Information storage & retrieval systems, Hadamard codes, Permutations, Cyclic codes, Markov spectrum
Abstract:	Quantum error correction lies at the heart of building reliable quantum information processing systems. Stabilizer codes, a fundamental class of quantum errorcorrecting codes, play a pivotal role in mitigating the adverse effects of noise and decoherence in quantum systems. This paper introduces a novel construction of quantum stabilizer codes using Hadamard difference sets, an elegant mathematical concept derived from combinatorial design theory. In this paper, the construction of the quantum stabilizer codes over non- cyclic Hadamard difference sets with parameters (4m²,2m²-m, m²-m), where m is a positive integer is discussed. Firstly, the parity check matrices are constructed from the Circulant permutation matrices with the help of Hadamard difference sets and then, the Symplectic inner product condition for Hadamard difference sets over binary operation for parity check matrices are obtained to affirm the commutative condition for Stabilizer operators which is vital for the error detection. For application, we constructed a Hadamard difference sets with parameters (16,6,2) for m = 2 of ordered pair of the group Z2 Z8 × (non-cyclic group) and quantum stabilizer codes are obtained by parity-check matrix. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

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Abstract:	Quantum error correction lies at the heart of building reliable quantum information processing systems. Stabilizer codes, a fundamental class of quantum errorcorrecting codes, play a pivotal role in mitigating the adverse effects of noise and decoherence in quantum systems. This paper introduces a novel construction of quantum stabilizer codes using Hadamard difference sets, an elegant mathematical concept derived from combinatorial design theory. In this paper, the construction of the quantum stabilizer codes over non- cyclic Hadamard difference sets with parameters (4m²,2m²-m, m²-m), where m is a positive integer is discussed. Firstly, the parity check matrices are constructed from the Circulant permutation matrices with the help of Hadamard difference sets and then, the Symplectic inner product condition for Hadamard difference sets over binary operation for parity check matrices are obtained to affirm the commutative condition for Stabilizer operators which is vital for the error detection. For application, we constructed a Hadamard difference sets with parameters (16,6,2) for m = 2 of ordered pair of the group Z2 Z8 × (non-cyclic group) and quantum stabilizer codes are obtained by parity-check matrix. [ABSTRACT FROM AUTHOR]
ISSN:	19929978