DECODING M-ARY LINEAR CODES WITH THE QUANTUM APPROXIMATE OPTIMIZATION ALGORITHM

Abstract The NP-hardness of the Minimum Distance Decoding Problem (MDDP) is the core of the McEliece cryptosystem. The difficulty of decoding a received word to the closest codeword in a given arbitrary code is key to its security. Related to the MDDP is the Coset Leader Problem (CLP), which consists in finding a word of a given syndrome and minimum Hamming weight. Both can be modelled as optimization problems, and solved using the Quantum Approximate Optimization Algorithm (QAOA), a well-known hybrid quantum- classical algorithm. In this paper, we model both the MDDP and CLP for linear codes over arbitrary m−ary alphabets, we make the theoretical analysis of the first level for the binary CLP problem, and introduce some experiments to test its performance. The experiments were carried out on both quantum computer simulators and real quantum devices, and use codes of different lengths and different depths of the QAOA.