A real-space Renormalization Group for Quantum Walks

A renormalization group (RG) treatment of quantum walks (QW) holds significant promise for insights into quantum transport phenomena and search algorithms in quantum computing. A key questions concerns universality, for instance, in the scaling properties of (unitary) quantum evolution depending on lattice type. Is there a single exponent describing the mean-square displacement (MSD) of QW, similar to the scenario observed in ordinary random walks, or is there a spectrum of modes, each with their own exponent? Does quantum interference ensure that these exponents are always smaller than for RW? To what extend does the lattice structure matter? For instance, are there differences between fractals and (translationally invariant!) hypercubic lattices, or even between triangular and square lattices within the same dimension? Most intriguing, how does the freedom in defining the quantum coin affect the scaling, if at all? It is the purpose of the RG to reveal such sweeping insights and provide classification (if not explanation) of regimes with potentially distinct behaviors.