A PromiseBQP-complete String Rewriting Problem

Abstract

We are given three strings s, t, and t' of length L over some fixed finite alphabet and an integer m that is polylogarithmic in L. We have a symmetric relation on substrings of constant length that specifies which substrings are allowed to be replaced with each other. Let Delta(n) denote the difference between the numbers of possibilities to obtain t from s and t' from s after n replacements. The problem is to determine the sign of Delta(m). As promises we have a gap condition and a growth condition. The former states that |Delta(m)| >= epsilon c^m where epsilon is inverse polylogarithmic in L and c>0 is a constant. The latter is given by Delta(n) <= c^n for all n. We show that this problem is PromiseBQP-complete, i.e., it represents the class of problems which can be solved efficiently on a quantum computer.