A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing

TL;DR

This paper explores how geometric algebra provides a unified, algebraic framework for describing quantum states, gates, and universality, offering conceptual clarity and computational benefits over traditional methods.

Contribution

It introduces a geometric algebra approach to quantum states and gates, and reexamines quantum gate universality using this formalism, unifying complex spaces with multivectors.

Findings

MSTA offers an explicit algebraic description of quantum states and gates.

GA formalism unites qubit space and unitary operators as multivectors.

Rotor-based rotations in GA are computationally advantageous.

Abstract

We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic configuration space), we present an explicit algebraic description of one and two-qubit quantum states together with a MSTA characterization of one and two-qubit quantum computational gates. Second, using the above mentioned characterization and the GA description of the Lie algebras SO(3) and SU(2) based on the rotor group Spin+(3, 0) formalism, we reexamine Boykin's proof of universality of quantum gates. We conclude that the MSTA approach does lead to a useful conceptual unification where the complex qubit space and the complex space of unitary operators acting on them become united, with both being made just by multivectors in real space. Finally, the GA approach to rotations based on the rotor group does bring conceptual and computational advantages compared to standard vectorial and matricial approaches.