A quantum algorithm for syndrome decoding of classical error-correcting linear block codes

Abstract

Decoding of linear block codes using syndrome decoding and finding whether there exists a codeword of weight $w$ in a set of codewords are known to belong to a class of NP-complete problems. For syndrome decoding of linear block codes, we provide an algorithm that gives a probabilistic solution with least Hamming weight. The algorithm can speed up this process significantly with the help of quantum search algorithm. For this, we make use of a search algorithm inspired from Grover's search. We also suggest a general approach for building a quantum circuit for syndrome detection. We test the algorithm in IBM Quantum Experience using qasm simulator and perform the complexity analysis identifying the algorithm's domain of utility.