Quantum Gates and Clifford Algebras

TL;DR

This paper explores the mathematical properties of Clifford algebras and their relevance to quantum gates, emphasizing their algebraic structure, matrix representations, and applications in quantum information science.

Contribution

It demonstrates how Clifford algebras provide a natural framework for representing quantum gates and linear transformations, facilitating quantum circuit modeling.

Findings

Cl(2n,C) includes all 2^n x 2^n complex matrices.

Clifford algebra structure simplifies modeling of quantum systems.

Linear operators in the algebra can represent quantum transformations.

Abstract

Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical properties of the algebras those also justifies such applications. First, any complex Clifford algebra with 2n generators, Cl(2n,C), has representation as algebra of all 2^n x 2^n complex matrices and so includes unitary matrix of any quantum n-gate. An arbitrary element of whole algebra corresponds to general form of linear complex transformation. The last property is also useful because linear operators are not necessary should be unitary if they used for description of restriction of some unitary operator to subspace. The second advantage is simple algebraic structure of Cl(2n) that can be expressed via tenzor product of standard "building units" and similar with behavior of composite quantum systems. The compact notation with 2n generators also can be used in software for modeling of simple quantum circuits by modern conventional computers.