Renormalisation of Quantum Cellular Automata

Quantum Cellular Automata (QCA) provide a general framework for local, discrete-time dynamics: a locality-preserving update acts uniformly on a lattice, enforcing finite information speed and well-defined causal structure. This dual viewpoint—as dynamical systems with causal cones and as parallel algorithms on distributed registers—supports applications across information theory and computation, many-body physics, and quantum field theory. Yet connecting microscopic update rules to emergent large-scale behavior remains subtle. This thesis develops tools to extract such large-scale features for QCA and their fermionic counterparts, Fermionic Cellular Automata (FCA), via two main advances. First, we close a long-standing gap in the circuit characterization of FCA. Building on the notion of stable equivalence, we prove that ancillary degrees of freedom are not fundamental: stable equivalence implies equivalence by a finite-depth circuit of local gates (F-equivalence). In one dimension, we obtain the fermionic analogue of the canonical QCA decomposition: an explicit two-layer implementation using size-4 blocks (a 4-Margolus partition), providing an operational bridge from graded automorphisms to concrete circuit architectures. Second, we introduce a Kadanoff-inspired renormalization procedure for Quantum and Fermionic Cellular Automata. By grouping cells and selecting suitable subspaces, multiple microscopic steps induce a single effective step on a coarse lattice. We derive necessary and sufficient algebraic conditions for renormalizability in arbitrary dimension and give an operator-level, easily testable criterion in 1D for a two-cell, two-step coarse-graining. A complete solution for qubit QCA and single-mode FCA shows that only dynamics with trivial information propagation are exactly renormalizable in these minimal settings; graded structures mildly enlarge admissible coarse-grainings but remain tightly constrained. These results clarify the rigidity of exact renormalization and motivate larger blocks, higher dimensions, or controlled approximations, with the conjecture that increasing on-site Hilbert space reveals nontrivial fixed points.