Group theory for quantum gates and quantum coherence
Michel Planat, Philippe Jorrand · Journal of Physics A Mathematical and Theoretical · 2008
Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group and stabilizing gates within an extension group . In this communication we explore the nice adequacy between group theoretical concepts such as commutators, normal subgroups, groups of automorphisms, short exact sequences, wreath products etc. and the coherent quantum computational primitives. The structure of the single-qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, we discover that M 20 , the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. We recover similar results by looking at the automorphisms of a complete set of mutually unbiased bases.