Mathematical Foundations of Holonomic Quantum Computer

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 higher-dimensional holonomies must be used to prove a universality of our Holonomic Quantum Computation.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 higher-dimensional holonomies must be used to prove a universality of our Holonomic Quantum Computation.

arXiv:quant-ph/0101102Jan 22, 20013 citations

More on Optical Holonomic Quantum Computer

Kazuyuki Fujii

We in this paper consider a further generalization of the (optical) holonomic quantum computation proposed by Zanardi and Rasetti (quant-ph/9904011), and reinforced by Fujii (quant-ph/9910069) and Pachos and Chountasis (quant-ph/9912093). We construct a quantum computational bundle on some parameter space, and calculate non-abelian Berry connections and curvatures explicitly in the special cases. Our main tool is unitary coherent operators based on Lie algebras su(n+1) and su(n,1), where the case of n = 1 is the previous one.We in this paper consider a further generalization of the (optical) holonomic quantum computation proposed by Zanardi and Rasetti (quant-ph/9904011), and reinforced by Fujii (quant-ph/9910069) and Pachos and Chountasis (quant-ph/9912093). We construct a quantum computational bundle on some parameter space, and calculate non-abelian Berry connections and curvatures explicitly in the special cases. Our main tool is unitary coherent operators based on Lie algebras su(n+1) and su(n,1), where the case of n = 1 is the previous one.

arXiv:quant-ph/0005129May 31, 20009 citations

Control aspects of holonomic quantum computation. (arXiv:quant-ph/0202055v4 UPDATED)

Dennis G. Lucarelli

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.

Mar 28, 20180 citations

Control aspects of holonomic quantum computation

Dennis Lucarelli

arXiv:quant-ph/0202055Journal of Mathematical PhysicsApr 6, 200515 citations

Holonomic quantum computation

Paolo Zanardi A

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 $n$ -fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold $\\cal M$ . The point of $\\cal M$ represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control manifold $\\cal M$ 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 $\\cal M.$ 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 $n$-fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold $\\cal M$. The point of $\\cal M$ represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control manifold $\\cal M$ 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 $\\cal M.$

Cambridge University Press eBooksSep 5, 20131 citation

Non-Abelian Berry connections for quantum computation

Jiannis K. Pachos, Paolo Zanardi A, Mario Rasetti

In the holonomic approach to quantum computation, information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-Abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible for a specific model to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multipartite system unitary transformations can be implemented efficiently by sequences of local holonomic gates. Moreover, a conceptual scheme for obtaining the required Hamiltonian family, based on frequently repeated pulses, is discussed, together with a possible process whereby the initial state can be prepared and the final one can be measured.In the holonomic approach to quantum computation, information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-Abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible for a specific model to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multipartite system unitary transformations can be implemented efficiently by sequences of local holonomic gates. Moreover, a conceptual scheme for obtaining the required Hamiltonian family, based on frequently repeated pulses, is discussed, together with a possible process whereby the initial state can be prepared and the final one can be measured.

arXiv:quant-ph/9907103Physical Review ADec 13, 1999316 citations

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

n

-fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold

\\cal M

. The point of

\\cal M

represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control manifold

\\cal M

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

\\cal M.