A Quantum Approach to the Unique Sink Orientation Problem

TL;DR

This paper explores quantum algorithms for the unique sink orientation problem on cubes, showing that efficient evaluation of certain functions leads to polynomial-time quantum solutions, connecting complexity theory and quantum computing.

Contribution

It introduces a reduction linking the evaluation of the kth power of the outmap to a polynomial-time quantum algorithm for the problem.

Findings

Quantum algorithms can solve the unique sink orientation problem efficiently if the outmap's kth power is efficiently evaluable.

The problem's complexity is connected to the ability to evaluate the outmap's powers.

Provides a new approach to tackling an intermediate complexity problem using quantum computation.

Abstract

We consider quantum algorithms for the unique sink orientation problem on cubes. This problem is widely considered to be of intermediate computational complexity. This is because there no known polynomial algorithm (classical or quantum) from the problem and yet it arrises as part of a series of problems for which it being intractable would imply complexity theoretic collapses. We give a reduction which proves that if one can efficiently evaluate the kth power of the unique sink orientation outmap, then there exists a polynomial time quantum algorithm for the unique sink orientation problem on cubes.