Observable-geometric phases and quantum computation

TL;DR

This paper introduces observable-geometric phases as a new approach to quantum phases, linking them to quantum geometry of observable space, and demonstrates their application in realizing universal quantum gates.

Contribution

It defines observable-geometric phases, connects them to quantum geometry, and applies them to implement universal quantum gates beyond traditional state-space methods.

Findings

Observable-geometric phases are linked to the quantum geometry of observable space.

These phases can be used to realize a universal set of quantum gates.

The scheme extends geometric quantum computation beyond state space geometry.

Abstract

This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. This scheme leads to the same gates as the Abelian geometric gates of Zhu and Wang \cite{ZW2002,ZW2003}, but based on the quantum geometry of the observable space beyond the state space.