A promiseBQP-complete string rewriting problem

We consider the following combinatorial problem. 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 ∆(n) denote the difference between the numbers of possibilities to obtain t from s and t from s after n ∈ N replacements. The problem is to determine the sign of ∆(m). As promises we have a gap condition and a growth condition. The former states that |∆(m)| ≥ ǫ cm where ǫ is inverse polylogarithmic in L and c > 0 is a constant. The latter is given by ∆(n) ≤ cn 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.