A promiseBQP-complete string rewriting problem

Abstract

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 $\Delta (n)$ denote the difference between the numbers of possibilities to obtain $t$ from $s$ and $t'$ from $s$ after $n \in\N$ 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 $|\Delta (m)| \geq \epsilon\,c^m$ where $\epsilon$ is inverse polylogarithmic in $L$ and $c>0$ is a constant. The latter is given by $\Delta (n) \leq c^n$ 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.