There have been approaches to finding quantum cellular automata inspired by elementary cellular automata in the past.
However, a method to create a quantum analogue for arbitrary elementary cellular automata is still missing.

The Wolfram code is a system for labeling all 256 classical elementary cellular automata.
This thesis presents a method to create a classically inspired quantum analogue for each elementary cellular automaton, in the form of a 1-dimensional spin-1/2 chain governed by a sum of local Hamiltonians.

Quantum analogues to non-reversible classical CAs are presented as non-unitary quantum systems governed by non-hermitian Hamiltonians.
Two possible sets of non-hermitian operators for governing such evolutions are presented and analyzed.

The resulting quantum systems are analyzed and compared to their classical counterparts for selected rules.
Similarities include convergence to a similar state, and showing the same behavior in static and periodic cases.
Known quantum phenomena like ergodicity and its breaking due to Hilbert space fragmentation are identified and explained for selected systems, where parallels to well-studied systems like the PXP-model are drawn.