Quantum Computational Riemannian and Sub-Riemannian Geodesics

K. Shizume, T. Nakajima, Ryo Nakayama, Y. Takahashi
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引用次数: 8

Abstract

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.
量子计算黎曼和次黎曼测地线
Nielsen等人在n量子位酉算子空间上引入了一个黎曼度量[M]。A. Nielsen, M. R. Dowling, M. Gu, A. C. Doherty, Science(2006), 1133。连接单位I和由这个度规定义的期望的酉算子W的最短曲线的长度本质上等于W的量子门复杂度。这个度规,因此,这个度规的黎曼测地线方程,有一个参数q叫做“惩罚”,它必须足够大。本文研究了将黎曼测地线方程的极限取为q→∞而得到的黎曼测地线方程,并证明了有限q的黎曼测地线可以由黎曼测地线显式构造。基于最优控制理论中的Krotov方法,提出了一种求连接I和W的(次)黎曼测地线的数值算法。作为一个例子,我们给出了一个连接I和被控-被控- z门的精确亚黎曼测地线,这是由算法的数值结果猜测得到的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Progress of Theoretical Physics
Progress of Theoretical Physics 物理-物理:综合
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