A Computational Toy Model of Non-Local Information Storage in a Quantum Cellular Automaton

Abstract

The holographic principle, which posits that the physics of a volume of space can be encoded on its boundary, has been profoundly reinterpreted as a form of quantum error correction. Separately, Quantum Cellular Automata (QCAs) offer a bottom-up model for the emergence of complexity. This paper proposes and tests the “Holographic QCA Hypothesis”: that complex states generated by “Goldilocks” QCAs naturally exhibit holographic non-local information storage. Using a direct state vector simulation of a 1D QCA ($N=12$), we demonstrate that a system evolved under the Fredkin gate generates high entanglement entropy and maintains robust mutual information between endpoints even after 75% of the intermediate chain is erased. This contrasts sharply with a control simulation using SWAP gates, where correlations collapse immediately. While strictly limited to a “toy model” regime by the exponential memory cost of classical simulation ($2^N$), these results provide a computational proof-of-principle that local unitary dynamics can spontaneously generate error-correcting properties, unifying concepts from complexity theory and holographic emergence.