A PromiseBQP-complete String Rewriting Problem

TL;DR

This paper proves that a specific string rewriting problem, with certain constraints and promises, is complete for the PromiseBQP class, indicating it can be efficiently solved by quantum computers.

Contribution

It establishes the PromiseBQP-completeness of a string rewriting problem under specific gap and growth conditions.

Findings

The problem is PromiseBQP-complete.

It characterizes the problem as solvable efficiently on quantum computers.

Provides a new link between string rewriting and quantum complexity classes.

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.