Portal de Periódicos da CAPES

$\mathbb {F}_p$-Linear and $\mathbb {F}_{p^m}$-Linear Qudit Codes From Dual-Containing Classical Codes

2021; Institute of Electrical and Electronics Engineers; Volume: 2; Linguagem: Inglês

10.1109/tqe.2021.3078152

2689-1808

Priya J. Nadkarni, Shayan Srinivasa Garani,

Quantum-Dot Cellular Automata

Quantum code construction from two classical codes D 1 [n,k 1 ,d 1 ] and D 2 [n,k 2 ,d 2 ] over the field F p m (p is prime and m is an integer) satisfying the dual containing criteria D 1 ⊥ ⊂ D 2 using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the F p -linear CSS codes and the well-known F p m -linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb F p -linear and F p m -linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For F p m -linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances d x and d z are equal to d 2 and d 1 , respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D 1 and D 2 , where one code is an F p l -linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the F p m -linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.