Application of Geometric Phase in Quantum Computations

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.

27 pages,minor corrections. This article is dedicated to memory of our dear friend, colleague and co-author Dr. Artur Tregubovich

References in corpus (22)

• Fault-tolerant quantum computation by anyons
• Implementation of universal quantum gates based on nonadiabatic geometric phases
• Non-adiabatic conditional geometric phase shift with NMR
• Geometric Quantum Computation
• Quantum Holonomies for Quantum Computing
• Quantum information and physics: some future directions
• Geometric nature of the environment-induced Berry phase and geometric dephasing
• Decoherence of geometric phase gates
• Holonomic quantum computation in the presence of decoherence
• Non-adiabatic geometrical quantum gates in semiconductor quantum dots
• Holonomic quantum gates: A semiconductor-based implementation
• Geometric phase shift in quantum computation using superconducting nanocircuits: nonadiabatic effects
• Proposal for measurment of harmonic oscillator Berry phase in ion traps
• Geometric phase in dephasing systems
• Realization of Arbitrary Gates in Holonomic Quantum Computation
• Exact Solutions of Holonomic Quantum Computation
• Use of non-adiabatic geometric phase for quantum computing by nuclear magnetic resonance
• Mixed state non-Abelian holonomy for subsystems
• Off-diagonal geometric phase in composite systems
• A Non-Adiabatic Controlled Not Gate for the Kane Solid State Quantum Computer
• Holonomic Quantum Computation Using Rf-SQUIDs Coupled Through A Microwave Cavity
• Evaluating holonomic quantum computation: beyond adiabatic limitation