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.

Highlights

  • We are concerned with finding an efficient quantum algorithm for a problem that admits no known classical polynomial time algorithm despite considerable effort: the unique sink orientation problem on cubes

  • Our reduction opens an approach towards obtaining an efficient quantum algorithm for the unique sink orientation problem on cubes

  • Finding the smallest ball enclosing a set of balls would admit a polynomial time solution if there was a polynomial time algorithm for the USO problem [12]

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Summary

INTRODUCTION

We are concerned with finding an efficient quantum algorithm for a problem that admits no known classical polynomial time algorithm despite considerable effort: the unique sink orientation problem on cubes. In this problem, one is given a directed graph on a hypercube such that every face (subcube) of the hypercube admits a unique sink vertex, and the goal is to find the global unique sink of the entire cube. Our reduction opens an approach towards obtaining an efficient quantum algorithm for the unique sink orientation problem on cubes

BACKGROUND
UNIQUE SINK ORIENTATION PROBLEM
PROPERTIES OF USO’s
QUANTUM APPROACH
EXPONENTIALLY LONG PERIODS
THE MISSING PIECE
VIII. CONCLUSION
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