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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Aspects of quantum computational geodesics

Howard E. Brandt

Published Mar 24, 2009• 1 citation
Publisher

Abstract

The object of this paper is to mathematically investigate characteristics of a geodesic equation describing possible minimum complexity paths in the special unitary group manifold representing the unitary evolution of n qubits associated with a quantum computation. Simple solutions are elaborated for the case of three qubits.
DOI
10.1117/12.817331

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In the Riemannian geometry of quantum computation, the quantum evolution is described in terms of the special unitary group of n-qubit unitary operators with unit determinant. To elaborate on some aspects of the methodology, the generic Jacobi equation and lifted Jacobi equation, together with solutions on the group manifold, are explicitly derived. This is important for investigations of the global characteristics of geodesic paths in the group manifold, and the determination of optimal quantum circuits for carrying out a quantum computation.
A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.
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