# Quantum Speedup Based on Classical Decision Trees

## TL;DR

This paper extends a quantum query complexity bound based on classical decision trees to non-binary functions using non-binary span programs, providing new quantum algorithms for graph problems.

## Contribution

It generalizes existing bounds to non-binary functions and introduces new quantum query algorithms for graph problems using span programs.

## Key findings

- Quantum algorithms for topological sorting with O(n^{3/2}) queries.
- Quantum upper bound for maximum bipartite matching: O(n^{3/4}√m + n).
- Generalization of decision tree quantum bounds to non-binary functions.

## Abstract

Lin and Lin have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function $f:\{0,1\}^n\to[m]$ whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for $f$ which makes at most $O(\sqrt{GT})$ quantum queries where $T$ is the depth of the decision tree and $G$ is the maximum number of mistakes of the guessing algorithm. In this paper we give a simple proof of and generalize this result for functions $f:[\ell]^n \to [m]$ with non-binary input as well as output alphabets. Our main tool for this generalization is non-binary span program which has recently been developed for non-binary functions, and the dual adversary bound. As applications of our main result we present several quantum query upper bounds, some of which are new. In particular, we show that topological sorting of vertices of a directed graph $\mathcal{G}$ can be done with $O(n^{3/2})$ quantum queries in the adjacency matrix model. Also, we show that the quantum query complexity of the maximum bipartite matching is upper bounded by $O(n^{3/4}\sqrt m + n)$ in the adjacency list model.