Quantum Algorithms of Solving the Backtracking of One-dimensional   Cellular Automata
  In [Wolfram 1982; Wolfram 1983; Wolfram 2002], the backtracking of one-dimensional cellular automata is to find out which of the 2n possible initial configurations of width n evolve to a specific configuration. In this paper, in one-dimensional cellular automata for a specific configuration of width n, its unique initial configuration can be found by mean of the proposed quantum algorithm with polynomial quantum gates, polynomial quantum bits and the successful probability that is the same as that of Shor's quantum order-finding algorithm in [Shor 1994]. 
Weng-Long Chang, Mang Feng, Kawuu Weicheng Lin, Chih-Chiang Wang, and   Ju-Chin Chen
cs.DS nlin.CG