Prime Factorization of Clifford Operators and Stabilizer States

Abstract

The Clifford operators (including the discrete Heisenberg-Weyl operators) and the stabilizer states play a basic role in the stabilizer formalism of quantum error correction and fault-tolerant quantum computation. For prime dimensional systems, they are well understood. However, for composite dimensional systems, subtleties arise due to the number-theoretic features of the system dimensions, and it is desirable to investigate how the structures of the Clifford operators and stabilizer states depend on the factorization of the systems. In this work, we construct explicitly a unitary map to implement the decomposition of the Clifford operators in any composite dimensional system induced by the prime factorization of the system. As applications, we obtain factorizations of stabilizer states in any system, and present a formula of the cardinality of the pure stabilizer states. In particular, we come to the remarkable observation that although the cardinality of stabilizer states is an increasing function of the prime dimension d,  it is not so for general d.