Quantum Reed-Muller codes

A set of quantum error correcting codes based on classical Reed-Muller codes is described. The codes have parameters [[n, k, d]] = [[2r, 2r −C(r, t)− 2 ∑t−1 i=0 C(r, i), 2 t + 2t−1]]. The study of quantum information is currently stimulating much interest. Most of the basic concepts of classical information theory have counterparts in quantum information theory, and among these is the idea of an error correcting code. An error correcting code is a means of storing information (whether quantum or classical) in a set of bits (ie either qubits or classical bits) in such a way that the information can be extracted even after a subset of the bits has changed in an unknown way. Such codes are a fundamental part of the study of classical information channels. The possibility of quantum error correction was only recently discovered [1, 2]. Importantly, it was shown that efficient quantum codes exist for arbitrarily large amounts of quantum information [3, 4]. The word ‘efficient’ refers to the fact that the rate k/n of the code need not fall off as n increases, for a given ratio d/n, where d is the minimum distance of the code. This and other features makes quantum error correction the best prospect for enabling quantum information to be transmitted or stored with a small amount of error, and consequently the best prospect for controlling noise in a quantum information processor. The subject of quantum error correction may be considered to have two distinct parts. The first part is to show how to apply error correction in a physical situation, and the second is to find good quantum error correcting codes. This paper is concerned with the second part, that of finding codes. Following [5], I will use the notation [[n, k, d]] to refer to a quantum error correcting code for n qubits having 2 codewords and minimum distance d (previously