A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices

TL;DR

This paper presents a method to construct quantum stabilizer codes using classical linear codes and Butson Hadamard matrices, providing a constructive proof and exploring conditions for stabilizer code formation.

Contribution

It introduces a new construction technique for quantum stabilizer codes based on classical codes and specific Hadamard matrices, expanding the toolkit for quantum error correction.

Findings

Constructs quantum codes from classical codes and Butson Hadamard matrices.

Provides a constructive proof for the existence of these quantum codes.

Analyzes conditions under which the codes are stabilizer codes.

Abstract

In this paper, we give a constructive proof to show that if there exist a classical linear code C is a subset of F_q^n of dimension k and a classical linear code D is a subset of F_q^k^m of dimension s, where q is a power of a prime number p, then there exists an [[nm, ks, d]]_q quantum stabilizer code with d determined by C and D by identifying the stabilizer group of the code. In the construction, we use a particular type of Butson Hadamard matrices equivalent to multiple Kronecker products of the Fourier matrix of order p. We also consider the same construction of a quantum code for a general normalized Butson Hadamard matrix and search for a condition for the quantum code to be a stabilizer code.