Dynamical Localization for General Scattering Quantum Walks

We consider quantum walks defined on arbitrary infinite graphs, parameterized by a family of scattering matrices attached to the vertices. Multiplying each scattering matrix by an i.i.d. random phase, we obtain a random scattering quantum walk. We prove dynamical localization for random scattering walks in a large-disorder regime. The result is based on a relation between fractional moment estimates and eigenfunction correlators of independent interest, which we establish for general random unitary operators.

💡 Research Summary

The paper studies dynamical localization for a broad class of quantum walks—so‑called scattering quantum walks (SQWs)—defined on arbitrary infinite graphs. Each vertex carries a scattering matrix, and the authors introduce disorder by multiplying each scattering matrix with an independent random phase (one random phase per vertex). The main result is that, in a large‑disorder regime, these random SQWs exhibit dynamical localization: the probability of finding the walker far from its initial region remains exponentially small uniformly in time.

To achieve this, the authors first develop a general framework for random unitary operators on the edge Hilbert space (\ell^{2}(E)). They define eigenfunction correlators (ECs) (\mathcal Q(\psi,\phi;I)) as the total variation of the spectral measure associated with two normalized vectors and an open arc (I) of the unit circle. They prove that ECs share the key properties of their self‑adjoint counterparts (boundedness, relation to time‑averaged matrix elements, RAGE theorem, lower semicontinuity under weak convergence). A family of “consistent” random unitaries ({U_{F}}{F\in\mathcal F}) is introduced, together with boundary operators (T{E,F}=U_{E}-U_{F}\oplus U_{F^{c}}) that measure the coupling between a finite subgraph (F) and its complement. When the boundary operators converge strongly to zero as (F) exhausts the graph, the finite‑volume spectral measures converge weakly, and the ECs satisfy a lower‑semicontinuity property.

The central technical theorem (Theorem 1) links two probabilistic assumptions to a bound on ECs. The first assumption is a fractional‑moment estimate: for some exponent (s\in(0,1)) and constant (C_{s}), the expectation of (| \mathbf 1_{f}(U_{B}-z)^{-1}\mathbf 1_{f’}|^{s}) is uniformly bounded for all (z) inside the unit disk. The second is a spectral‑averaging bound: the expected norm of (\mathbf 1_{e},\mathrm{Re}\bigl((U+z)(U-z)^{-1}\bigr),\mathbf 1_{e}) is bounded by a constant (C_{W}). Under these hypotheses, for any (\beta\in(0,s)) the expected interpolated EC satisfies
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