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arXiv:1104.3771 · Journal of Physics Conference Series

Geometric phase and gauge theory structure in quantum computing

A Bruno, Antonio Capolupo, S Kak, G Raimondo, Giuseppe Vitiello

Published Jul 8, 2011• 2 citations
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Abstract

We discuss the presence of a geometrical phase in the evolution of a qubit state and its gauge structure. The time evolution operator is found to be the free energy operator, rather than the Hamiltonian operator.
DOI
10.1088/1742-6596/306/1/012065

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In this dissertation we discuss several aspects of a two level system (qubit) in the context of quantum mechanics and quantum eld theory. The presence of geometrical phases in the evolution of a qubit state is shown. We study geometric structures, which are correlated to an unitary time evolution and its interesting gauge structure. They can be very useful in quantum computational processes. We illustrate the quantum eld theoretical formulation of boson mixed elds, and oscillation formulas for neutral and charged elds are found. We show that the space for the mixed elds is unitary inequivalent to the state space where the unmixed eld are de ned, and we also derive the structure of the currents and charges for the charged mixed elds. Phenomenological aspects of meson mixing in the presence of the decay are discussed. In particular, we show that the the e ective Hamiltonian is non-Hermitian and non-normal in the Wigner-Weisskopf approximation and we use the biorthonormal basis formalism to diagonalize such an Hamiltonian. Finally, the presence of CP and CPT violations in meson mixing is shown.
Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2) , characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024