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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 26, No. 2, pp. 247-264 (2010)
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Quantum Computational Jacobi Fields

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Howard E. Brandt

U.S. Army Research Laboratory

**Abstract:** 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 several aspects of the methodology, the Riemannian curvature, geodesic equation, Jacobi equation, and lifted Jacobi equation 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.

**Keywords:** quantum computing, quantum circuits, quantum complexity, unitary group, differential geometry, Riemannian geometry, curvature, geodesics, Lax equation, Jacobi fields

**Classification (MSC2000):** 81P68, 81-01, 81-02, 53B20, 53B50, 22E60, 22E70, 03D15, 53C22; 22D10, 43A75, 51N30, 20C35, 81R05

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