Exact new-sector computation and finite-size spectral statistics of Grover walks on random graph lifts

Abstract

We study finite-size spectral statistics of the positive new eigenangles of untwisted Grover walks on random lifts of four fixed regular base graphs: K4, the Petersen graph, the Frucht graph, and K5. The determinant identity of Konno and Sato, the old/new decomposition of graph covers, the covering determinant decomposition, and the Kesten–McKay convergence theorem used here are previously published results. Our contribution is to put these ingredients into an explicit computational framework, document a stable blockwise algorithm for the compressed new sector, correct the finite-size unfolding normalization, and carry out a reproducible comparison across twelve family-size configurations. The study includes a non-vertex-transitive cubic base, a quartic base, and lifts up to sheet number N=200. Nearest-neighbor spacings are compared with descriptive Poisson and Wigner-surmise curves, while graph-level spacing ratios are calibrated against size-matched GOE simulations. The observed ratio values are much closer to the finite-size GOE calibration than to unitary or Poisson benchmarks. Graph-level Kolmogorov–Smirnov distributions, pooled empirical-unfolding checks, and connectivity diagnostics are also reported. The results provide finite-size evidence for orthogonal-type local statistics in the selected lift families; they do not constitute a proof of GOE universality.