On group theory for quantum gates and quantum coherence

Michel Planat (1) , Philippe Jorrand (2)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
2 CAPP - Calculs algorithmes programmes et preuves

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 P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one discovers that M20, the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. One recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases.

Domaines: Physique Quantique [quant-ph]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Théorie des groupes [math.GR]

Michel Planat, Philippe Jorrand. On group theory for quantum gates and quantum coherence. Journal of Physics A: Mathematical and Theoretical, 2008, 41, pp.182001. ⟨hal-00263678v2⟩