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Control algebra for holonomic quantum computation with squeezed coherent states

Dennis G. Lucarelli

Published Feb 10, 2002• 2 citations
OpenAlex

Abstract

A condition for determining the holonomy group associated to a principal bundle with connection is proven. Within the context of Holonomic Quantum Computation, this condition aids in determining the universality of the system. We apply the theory to the product bundle arising from the full two-qubit system of holonomic computation with squeezed coherent states and conclude that the control algebra generates the Lie algebra u(4).

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Mathematical Foundations of Holonomic Quantum Computer

Kazuyuki Fujii

This is a sequel to the papers (quant-ph/9910063) and (quant-ph/0004102). The aim of this paper is to give mathematical foundations to Holonomic Quantum Computation (Computer) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093). In 2-qubit case we give an explicit form to non-abelian Berry connection of quantum computational bundle which is associated with Holonomic Quantum Computation, on some parameter space. We also suggest a possibility that not only usual holonomy but also higherdimensional holonomies must be used to prove a universality of our Holonomic

•Apr 27, 2000•63 citations
A condition for determining the holonomy group associated to a principal bundle with connection is proven. Within the context of Holonomic Quantum Computation, this condition aids in determining the universality of the system. We apply the theory to the product bundle arising from the full two-qubit system of holonomic computation with squeezed coherent states and conclude that the control algebra generates the Lie algebra u(4).
Chowaposaposaposs theorem and universal holonomic quantum computation
A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to holonomic quantum computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated with a non-Abelian gauge connection. When applied to the optical holonomic computer, these conditions determine that the holonomy group of the two-qubit interaction model contains SU (2) × SU (2). In particular, a universal two-qubit logic gate is attainable for this model.
A unifying framework for the control of quantum systems with non-Abelian holonomy is presented. It is shown that, from a control theoretic point of view, holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle. An extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm.
A unifying framework for the control of quantum systems with non-Abelian holonomy is presented. It is shown that, from a control theoretic point of view, holonomic quantum computation can be treated as a control system evolving on a principal fiber bundle. An extension of methods developed for these classical systems may be applied to quantum holonomic systems to obtain insight into the control properties of such systems and to construct control algorithms for two established examples of the computing paradigm.
We present a geometric framework for holonomic quantum computing in which quantum gates arise from global properties of control manifolds rather than fine-tuned dynamical evolution. Quantum states are modeled as complex projective fibers over a classical control manifold, and adiabatic loops induce unitary gates through Berry and Wilczek–Zee holonomy. Within this setting, we introduce _Quantum Inner State Manifolds_ (QISMs) as symplectic fiber bundles equipped with a natural unitary connection governed by the Fubini–Study form. Using the Ambrose–Singer theorem, we show that generic QISMs generate holonomy groups dense in U(N)U{(N)}U(N), establishing universality. Fault tolerance emerges from global geometric features, providing a robust geometric foundation for quantum gate design.
This is a sequel to the papers (quant-ph/9910063) and (quant-ph/0004102). The aim of this paper is to give mathematical foundations to Holonomic Quantum Computation (Computer) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093). In 2-qubit case we give an explicit form to non-abelian Berry connection of quantum computational bundle which is associated with Holonomic Quantum Computation, on some parameter space. We also suggest a possibility that not only usual holonomy but also higherdimensional holonomies must be used to prove a universality of our Holonomic