Renormalization of the Unitary Evolution Equation for Coined Quantum Walks

TL;DR

This paper develops a rigorous real-space renormalization group framework for analyzing coined quantum walks on fractal networks, extending classical RG methods to accurately predict quantum walk scaling behaviors.

Contribution

It introduces an extended RG fixed-point analysis incorporating subdominant eigenvalues, enabling analytical derivation of quantum walk dimensions on fractals.

Findings

Derived analytical expressions for quantum walk dimensions on fractals.

Extended RG analysis includes subdominant eigenvalues for accuracy.

Validated conjectured quantum walk scaling results on various fractal networks.

Abstract

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 $d_{w}^{QW}=\log_{2}\sqrt{\lambda_{1}\lambda_{2}}$. 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.