On The Complexity of Quantum Circuit Manipulation

The stabilizer class of circuits, introduced by Daniel Gottesman, consists of quantum circuits in which every gate is a controlled-NOT (CNOT), Hadamard or phase gate [5]. These circuits have several interesting properties. For example, they are robust enough to allow for entangled states, yet they are known to be simulable in polynomial time by a classical computer. These circuits also naturally arise in the study of quantum error-correcting codes. Iwama et. al. introduced a complete set of local transformations for CNOT circuits [7]. That is, they gave a list of transformation rules, each of which involve a constant number of gates, which suffice to take any CNOT circuit to any equivalent such circuit. Such sets of rules typically have applications in developing heuristic circuit minimization techniques. In previous work, we built upon these rules to give a more general set of local transformations for the more general class of stabilizer circuits. With these rules in hand, several questions naturally arise, which we address in this paper.