Geometric phase in the G3+ quantum state evolution

TL;DR

This paper explores how representing quantum states within geometric algebra enhances the understanding of their evolution and geometric phase, with implications for topological quantum computing.

Contribution

It introduces a geometric algebra framework for quantum states that explicitly defines the complex plane and improves the calculation of geometric phases.

Findings

Explicit geometric phase calculations are possible within the G3+ algebra framework.

The approach offers a more detailed description of quantum state evolution.

Implications for topological quantum computing are discussed.

Abstract

When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.