Christopher Jack Turner

We study spectral statistics and recurrences in translationally-invariant Clifford circuits built from brickwork layers of two-qubit entangling gates and single-qubit layers. We first analyze strictly Clifford circuits, whose spectra exhibit strong degeneracies rooted in stabilizer dynamics and finite-order structure. We then develop a powerful set of algorithms to compute Pauli operator orbits associating them with minimal polynomials over GF(2). Extending to Clifford+T circuits, we move beyond tableau simulation, compute operator Shannon entropy in the Pauli basis, and perform symmetry-resolved level-spacing analyses. We identify families mappable to free-fermion (TFIM-like) dynamics with Poisson statistics and incommensurate quasienergies, as well as circuits displaying fragmentation, near-integrable behavior, and GOE-like chaos when dihedral symmetry is broken or boundary conditions are modified. Our results provide a unified framework for diagnosing integrability, chaos, and scrambling in structured Clifford and non-Clifford Floquet architectures.