On Clifford groups in quantum computing

TL;DR

This paper explores the structure and symmetries of Clifford groups in quantum computing, emphasizing their mathematical properties and their relation to Weyl-Heisenberg groups across finite-dimensional quantum systems.

Contribution

It provides a comprehensive analysis of Clifford groups as quotients related to symmetries of Pauli gradings in finite quantum systems, extending understanding beyond multiqubit models.

Findings

Clifford groups leave Weyl-Heisenberg groups invariant.

Symmetries of Pauli gradings correspond to Clifford group structures.

Results apply to both single and composite finite quantum systems.

Abstract

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, $\pi/4$-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 $\U(1)$.