Quantum computing based on complex Clifford algebras

TL;DR

This paper introduces a novel representation of qubits and quantum gates using complex Clifford algebras, enabling straightforward Dirac formalism application and quantum computation demonstrations.

Contribution

It presents a new algebraic framework for quantum computing based on complex Clifford algebras, extending previous real algebra approaches.

Findings

Successfully performed quantum computations with standard quantum gates.

Compared complex and real Clifford algebra representations.

Showed the framework's potential for simplifying quantum formalism.

Abstract

We propose to represent both $n$--qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension $2n.$ In this framework, the Dirac formalism can be realized in straightforward way. We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates. We also compare our approach with representations that use real geometric algebras.