Quantum Reed-Solomon Codes

During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [8]) and finding pre– images of an n–ary Boolean function (“searching”) in time O( √ 2n) (see [5]). Quantum computers are not only of theoretical nature—there are several suggestions how to physically realize them (see, e. g., [2, 3]). On the way towards building a quantum computer, one very important problem is to stabilize quantum mechanical systems since they are very vulnerable. A theory of quantum error–correcting codes has already been established (see [6]). Nevertheless, the problem of how to encode and decode quantum error–correcting codes has hardly been addressed, yet. We present the construction of quantum error–correcting codes based on classical Reed–Solomon (RS) codes. For RS codes, many classical decoding techniques exist. RS codes can also be used in the context of erasures and for concatenated codes. Encoding and decoding of quantum RS codes is based on quantum circuits for the cyclic discrete Fourier transform over finite fields which are presented in the full paper, together with the quantum implementation of any linear transformation over finite fields. We start with a brief introduction to quantum computation and quantum error–correcting codes, followed by some results about binary codes obtained from codes over extension fields.