Analysis of Coined Quantum Walks with Renormalization

We introduce a new framework to analyze quantum algorithms with the renormalization group (RG). To this end, we present a detailed analysis of the real-space RG for discrete-time quantum walks on fractal networks and show how deep insights into the analytic structure as well as generic results about the long-time behavior can be extracted. The RG-flow for such a walk on a dual Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained explicitly from elementary algebraic manipulations, after transforming the unitary evolution equation into Laplace space. Unlike for classical random walks, we find that the long-time asymptotics for the quantum walk requires consideration of a diverging number of Laplace-poles, which we demonstrate exactly for the closed form solution available for the walk on a 1d-loop. In particular, we calculate the probability of the walk to overlap with its starting position, which oscillates with a period that scales as $N^{d_{w}^{Q}/d_{f}}$ with system size $N$. While the largest Jacobian eigenvalue $λ_{1}$ of the RG-flow merely reproduces the fractal dimension, $d_{f}=\log_{2}λ_{1}$, the asymptotic analysis shows that the second Jacobian eigenvalue $λ_{2}$ becomes essential to determine the dimension of the quantum walk via $d_{w}^{Q}=\log_{2}\sqrt{λ_{1}λ_{2}}$. We trace this fact to delicate cancellations caused by unitarity. We obtain identical relations for other networks, although the details of the RG-analysis may exhibit surprisingly distinct features. Thus, our conclusions -- which trivially reproduce those for regular lattices with translational invariance with $d_{f}=d$ and $d_{w}^{Q}=1$ -- appear to be quite general and likely apply to networks beyond those studied here.