#### Title

A geometric algebra approach ton-qubit systems

#### Date of Award

2007

#### Degree Type

Dissertation

#### Department

Physics

#### Rights

CC BY-NC-ND 4.0

#### Abstract

A geometric formalism for the treatment of n-qubit systems is presented in erms of the Clifford's Geometric algebra Cfm, as an alternative to the tra itional matrix formulation. This objective is accomplished by generalizing he one-qubit system formulated in terms of the bivector space of Cfg. This ormulation is based in the well known isomorphism between the so(3) and u(2) Lie algebras. It is known that a quantum system with N orthogonal states (levels) is ontrollable with the SU(N) group. However, a system with an even number of orthogonal states may also be state-controllable with the USp(N) group, hich is a subgroup of SU(N) with a Lie algebra isomorphic with the Lie lgebra of the symplectic group Sp(N). The isomorphism between the sp(4) and spin(5) Lie algebras allows the ormulation of a two-qubit system in terms of the the bivector space of Cfs, as a natural instance of the spin(5) Lie algebra. Another isomorphism be ween the sp( 4) Lie algebra and the anti-Hermitian space of Cf4 is revealed, herefore allowing the formulation of a two-qubit system in terms of Cf4 as ell. More isomorphisms are exposed between some subspaces of higher di ensional Clifford algebras with the Lie algebras of the USp(N) and SU(N) roups. The immediate consequence is the possibility to represent an n-qubit iv A geometric formalism for the treatment of n-qubit systems is presented in erms of the Clifford's Geometric algebra Cfm, as an alternative to the tra itional matrix formulation. This objective is accomplished by generalizing he one-qubit system formulated in terms of the bivector space of Cfg. This ormulation is based in the well known isomorphism between the so(3) and u(2) Lie algebras. It is known that a quantum system with N orthogonal states (levels) is ontrollable with the SU(N) group. However, a system with an even number of orthogonal states may also be state-controllable with the USp(N) group, hich is a subgroup of SU(N) with a Lie algebra isomorphic with the Lie lgebra of the symplectic group Sp(N). The isomorphism between the sp(4) and spin(5) Lie algebras allows the ormulation of a two-qubit system in terms of the the bivector space of Cfs, as a natural instance of the spin(5) Lie algebra. Another isomorphism be ween the sp( 4) Lie algebra and the anti-Hermitian space of Cf4 is revealed, herefore allowing the formulation of a two-qubit system in terms of Cf4 as ell. More isomorphisms are exposed between some subspaces of higher di ensional Clifford algebras with the Lie algebras of the USp(N) and SU(N) roups. The immediate consequence is the possibility to represent an n-qubit iv

#### Recommended Citation

Cabrera Lafuente, Renan Andres, "A geometric algebra approach ton-qubit systems" (2007). *Electronic Theses and Dissertations*. 4614.

https://scholar.uwindsor.ca/etd/4614