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 Δ(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 Δ(m). As promises we have a gap condition and a growth condition. The former states that |Δ(m)| ≥ ∈ cm where ∈ is inverse polylogarithmic in L and c \u3e 0 is a constant. The latter is given by Δ(n) ≤ cn for all n. We show that this problem is PromiseBQP-complete, i.e., it represents the class of problems that can be solved efficiently on a quantum computer. © Rinton Press