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arXiv:quant-ph/0202055 · Journal of Mathematical Physics

Control aspects of holonomic quantum computation

Dennis Lucarelli

Published Apr 6, 2005• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.1888028

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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.
We show that the notion of generalized Berry phase i.e., non-abelian holonomy, can be used for enabling quantum computation. The computational space is realized by a nnn-fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold calM\\cal McalM. The point of calM\\cal McalM represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control manifold calM\\cal McalM induce non trivial unitary transformations on the computational space. For a generic system it is shown that this mechanism allows for universal quantum computation by composing a generic pair of loops in calM.\\cal M.calM.
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