Design of Quantum Stabilizer Codes From Quadratic Residues Sets

Abstract

We propose two types, namely Type-I and Type-II, quantum stabilizer codes using quadratic residue sets of prime modulus given by the form $p=4n\pm 1$. The proposed Type-I stabilizer codes are of cyclic structure and code length $N=p$. They are constructed based on multi-weight circulant matrix generated from idempotent polynomial, which is obtained from a quadratic residue set. The proposed Type-II stabilizer codes are of quasi-cyclic (QC) structure and code length $N=pk$, where $k$ is the size of a quadratic residue set. They are constructed based on structured sparse-graphs codes derived from proto-matrix and circulant permutation matrix. With the proposed methods, we design rich classes of cyclic and quasi-cyclic quantum stabilizer codes with variable code length. We show how the commutative constraint (also referred to as the Symplectic Inner Product constraint) for quantum codes can be satisfied for each proposed construction method. We also analyze both the dimension and distance for Type-I stabilizer codes and the dimension of Type-II stabilizer codes. For the cyclic quantum stabilizer codes, we show that they meet the existing distance bounds in literature.

References in corpus (13)

• Mixed State Entanglement and Quantum Error Correction
• Good Quantum Error-Correcting Codes Exist
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• Quantum Error Correction and Orthogonal Geometry
• Stabilizer Codes and Quantum Error Correction
• Sparse Graph Codes for Quantum Error-Correction
• Codeword Stabilized Quantum Codes
• A nonadditive quantum code
• Convolutional and tail-biting quantum error-correcting codes
• A Proposed Quantum Low Density Parity Check Code
• A simple family of nonadditive quantum codes
• Constructions of Quantum Convolutional Codes
• A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices