Quantum Computational Riemannian and Sub-Riemannian Geodesics(General and Mathematical Physics)

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概要

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• Nielsen et al. have introduced a Riemannian metric on the space of n-qubit unitary operators [M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Science 311 (2006), 1133]. The length of the shortest curve connecting the identity I and a desired unitary operator W defined by this metric is essentially equivalent to the quantum gate complexity of W. This metric, and thus, the Riemannian geodesic equation for this metric, has a parameter q called "penalty" that must be large enough. In this paper, we investigate the sub-Riemannian geodesic equation obtained by taking the limit of the Riemannian geodesic equation as q→∞, and show that the Riemannian geodesics for finite q can be explicitly constructed from the sub-Riemannian geodesics. We also present a numerical algorithm for finding the (sub-)Riemannian geodesics connecting I and W, which is based on the Krotov method in optimal control theory. As an example, we give an exact sub-Riemannian geodesic connecting I and the controlled-controlled-Z gate, which is obtained by guessing from the numerical results of the algorithm.

• 2012-06-25

著者

• Takahashi Yutaka

  Graduate School Of Informatics Kyoto University

• Takahashi Yutaka

  Graduate School Of Science And Engineering Yamagata University

• SHIZUME Kosuke

  Graduate School of Library, Information and Media Studies, University of Tsukuba

• NAKAJIMA Takao

  Graduate School of Library, Information and Media Studies, University of Tsukuba

• NAKAYAMA Ryo

  College of Knowledge and Library Science, School of Informatics, University of Tsukuba

• Nakayama Ryo

  College Of Knowledge And Library Science School Of Informatics University Of Tsukuba

• Shizume Kosuke

  Graduate School Of Library Information And Media Studies University Of Tsukuba

• Nakajima Takao

  Graduate School Of Library Information And Media Studies University Of Tsukuba

• TAKAHASHI Yutaka

  Graduate School of Science and Engineering, Yamagata University

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