Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

TL;DR

This paper introduces a quantum algorithm for dynamic programming on DAGs, achieving a quadratic speedup over classical methods for problems like Boolean formula evaluation, longest path, and diameter search.

Contribution

The paper presents a novel quantum dynamic programming algorithm for DAG problems, improving computational efficiency for specific applications.

Findings

Quantum algorithm runs in O(√(n̂ m) log n̂) time.

Applicable to problems using OR, AND, NAND, MAX, MIN functions.

Provides speedup for Boolean formula evaluation and path problems on DAGs.

Abstract

In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluating. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.