Quantum Cellular Automata, Black Hole Thermodynamics, and the Laws of Quantum Complexity

Abstract

This paper introduces a new formalism for quantum cellular automata (QCAs), based on evolving tensor products of qubits using local unitary operators. It subsequently uses this formalism to analyze and validate several conjectures, stemming from a formal analogy between quantum computational complexity theory and classical thermodynamics, that have arisen recently in the context of black hole physics. In particular, the apparent resonance and thermalization effects present within such QCAs are investigated, and it is demonstrated that the expected exponential relationships between the quantum circuit complexity of the evolution operator, the classical entropy of the equilibrium QCA state, and the characteristic equilibration time of the QCA, all hold within this new model. Finally, a rigorous explanation for this empirical relationship is provided, as well as for the relationship with black hole thermodynamics, by drawing an explicit mathematical connection with the mean ergodic theorem, and the ergodicity of k-local quantum systems.