Application of Geometric Phase in Quantum Computations

TL;DR

This paper explores how geometric phases can be utilized to implement quantum gates in holonomic quantum computers, providing explicit formulas for both adiabatic and non-adiabatic cases, and demonstrating the potential for universal quantum computation.

Contribution

It introduces explicit expressions for quantum gates based on geometric phases in holonomic quantum computing, including both Abelian and non-Abelian phases, and discusses their application in different physical models.

Findings

Explicit formulas for one- and multi-qubit gates using geometric phases.

Demonstration of universal quantum gate construction in specific models.

Analysis of non-adiabatic geometric phases for quantum computation.

Abstract

Geometric phase that manifests itself in number of optic and nuclear experiments is shown to be a useful tool for realization of quantum computations in so called holonomic quantum computer model (HQCM). This model is considered as an externally driven quantum system with adiabatic evolution law and finite number of the energy levels. The corresponding evolution operators represent quantum gates of HQCM. The explicit expression for the gates is derived both for one-qubit and for multi-qubit quantum gates as Abelian and non-Abelian geometric phases provided the energy levels to be time-independent or in other words for rotational adiabatic evolution of the system. Application of non-adiabatic geometric-like phases in quantum computations is also discussed for a Caldeira-Legett-type model (one-qubit gates) and for the spin 3/2 quadrupole NMR model (two-qubit gates). Generic quantum gates for these two models are derived. The possibility of construction of the universal quantum gates in both cases is shown.