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Renormalization of the unitary evolution equation for coined quantum walks

机译：成型量子宽度的单一演化方程的重整化

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We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep insights into the relation between classical and quantum walks, its analytic basis has remained obscure. Our discussion here is laying the groundwork for a rigorous implementation of the RG for this important class of transport and algorithmic problems, although some instances remain unresolved. Specifically, we find that the RG fixed-point analysis of the classical walk, which typically focuses on the dominant Jacobian eigenvalue lambda(1), with walk dimension d(w)(RW)= log(2) lambda(1), needs to be extended to include the subdominant eigenvalue lambda(2), such that the dimension of the quantum walk obtains d(w)(QW) = log(2)root lambda(1) lambda(2). With that extension, we obtain analytically previously conjectured results for d(w)(QW) of Grover walks on all but one of the fractal networks that have been considered.
机译：我们考虑离散时间的演化方程，其中经典随机步行的随机运营商被整体操作员代替。这样的问题是作为创造量子行走的框架，这是对物理可实现的，低维几何形状中的量子搜索算法的格罗弗限制至关重要的框架。特别是，我们分析了最近引入的确切实际空间重整组（RG）程序研究了在分形网络上的量子展开量级。虽然该程序在数字实施时，能够对古典和量子漫游之间的关系提供一些深入的见解，但其分析基础仍然模糊不清。我们的讨论正在为这类重要的运输和算法问题奠定RG的严格实施的基础，尽管某些情况仍未解决。具体而言，我们发现古典散步的RG定点分析，通常专注于主导的雅碧碧眼度量λ（1），步行尺寸D（W）（RW）= log（2）Lambda（1），需要扩展到包括次侏儒特征值λ（2），使得量子步行的尺寸获得D（w）（qw）= log（2）根Lambda（1）λ（2）。通过这种延伸，我们获得了先前获得的Grover D（W）（QW）的经过分析猜想的结果，除了考虑的一条分形网络之外。

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