The Problem of Dynamic Programming on a Quantum Computer

TL;DR

This paper explores the limits of quantum computing for solving finite-horizon dynamic programming problems, establishing bounds on potential speedups and introducing a query model for analyzing quantum algorithms.

Contribution

It introduces a query model for quantum DP algorithms, provides oracle constructions for key problems, and proves lower bounds limiting quantum speedups for DP.

Findings

No greater-than-quadratic quantum speedup for DP problems

Quantum algorithms have quadratic speedup limits for DP

Lower bounds established for classical and quantum DP algorithms

Abstract

We discuss the problem of finite-horizon dynamic programming (DP) on a quantum computer. We introduce a query model for studying quantum and classical algorithms for solving DP problems, and provide example oracle constructions for the travelling salesperson problem, the minimum set-cover problem, and the edit distance problem. We formulate open questions regarding quadratic quantum speedups for DP and discuss their implications. We then prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for DP, and show that no greater-than-quadratic speedup can be achieved for solving DP problems.