A decision procedure for unitary linear quantum cellular automata

TL;DR

This paper presents an efficient algorithm to determine whether a linear quantum cellular automaton is unitary, ensuring it adheres to the fundamental constraints of quantum mechanics, with a complexity of O(n^3).

Contribution

It introduces a novel, efficient decision procedure for checking unitarity in linear quantum cellular automata, a key requirement for their validity in quantum computing.

Findings

Algorithm runs in O(n^3) time for automata with neighborhood size r.

Provides a practical method to verify unitarity in quantum cellular automata.

Enhances the theoretical foundation for quantum automata validation.

Abstract

Linear quantum cellular automata were introduced recently as one of the models of quantum computing. A basic postulate of quantum mechanics imposes a strong constraint on any quantum machine: it has to be unitary, that is its time evolution operator has to be a unitary transformation. In this paper we give an efficient algorithm to decide if a linear quantum cellular automaton is unitary. The complexity of the algorithm is O(n^((3r-1)/(r+1))) = O(n^3) in the algebraic computational model if the automaton has a continuous neighborhood of size r, where $n$ is the size of the input.