Exact localization of Grover quantum walks on crystal lattices: transfer currents, a universal bound, and uniform-spanning-tree degree laws

Abstract

Exact results for the localization ("trapping") of discrete-time Grover quantum walks on crystal lattices. A finite fraction of a Grover walker stays at its starting point forever, because the walk operator has flat eigenvalues. This paper computes the long-time return probability p̄ exactly for many lattices. Main results:• For the flip-flop Grover walk, the +1 and −1 flat-band projectors are I − Y and I − Y^Q, where Y and Y^Q are the Kirchhoff transfer-current matrices of the Laplacian and signless Laplacian. This reduces trapping to lattice Green functions. It gives closed forms in terms of Watson integrals (simple cubic, BCC, FCC), complete elliptic integrals (triangular, kagome, checkerboard, star, line graphs) and elementary functions of the bond anisotropy.• A universal lower bound p̄ ≥ (d−2)² / [2d(d−1)] for edge-transitive lattices of vertex degree d, attained exactly on the honeycomb, diamond and Laves graph lattices.• Exact trapping for lazy (Szegedy) walks on the square, triangular, honeycomb and diamond lattices as functions of the laziness.• For the moving-shift walk on Z^d, trapping is governed by A_d = ∫₀^∞ erfc(√s)^d ds, with elementary closed forms for d ≤ 5.• The degree of a vertex in the uniform spanning tree is exactly 1 + Binomial(z−1, 1/(z−1)) on 2-arc-transitive lattices, with closed-form degree distributions for several other lattices. All closed forms were identified by integer-relation (PSLQ) searches at 25–80 digits and checked by Brillouin-zone diagonalization and direct simulation of the walks. AI-use disclosure: this research, including choosing the problems, discovering the formulas, running all computations, and writing the manuscript, was carried out with extensive use of an AI system (Claude Code, by Anthropic) under the author's direction. It has not been peer reviewed. The author takes full responsibility for the content. Code: https://github.com/JacobGoodchild/findformula