On Clifford groups in quantum computing

J Tolar

doi:10.1088/1742-6596/1071/1/012022

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, 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).

Highlights

In quantum mechanics of single N -level systems in Hilbert spaces of finite dimension N, the basic operators are the generalized Pauli matrices
Its normalizer within the unitary group U(N ), or in other words the largest subgroup of the unitary group having the Weyl-Heisenberg group as a normal subgroup, is in the papers on quantum information conventionally called the Clifford group [5]. Since this normalizer necessarily contains the continuous group U(1) of phase factors, some authors adopt an alternative definition of the Clifford group as the quotient of the normalizer with respect to U(1) [6]
Our results on symmetries of the Pauli gradings of quantum operator algebras [7, 8, 9] account for all possible Clifford quotient groups corresponding to arbitrary single or composite quantum systems

Summary

Introduction

In quantum mechanics of single N -level systems in Hilbert spaces of finite dimension N , the basic operators are the generalized Pauli matrices. 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).

Results

Conclusion