Clifford meets Clifford in Quantum Computing: On Clifford seminearrings and Clifford groups

D. M. Appleby, Bengtsson, I., Brierley, S. +3 more

Quantum Information and Computation

Miroslav Korbelář, J. Tolar

Abstract The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space Z N ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of S L ( 2 , Z N ) it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

Thalanayar S. Santhanam

J. Tolar

The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π /4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation. In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).

M. Gregorič, N. S. Mankoč Borštnik

Michel R. P. Planat

One reports about the embedding of the Pauli group into the Clifford group, and one explicits the concept of Clifford quantum computation, following Gottesman-Knill approach. Normal subgroups of the two-qubit Clifford group are investigated and a link to the generalized quadrangle of other two and hexads in the Steiner system S(3,6,22) is found.