Quantum Speedup Based on Classical Decision Trees

Salman Beigi,Leila Taghavi

doi:10.22331/q-2020-03-02-241

Abstract

Lin and Lin \cite{LL16} 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→[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(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:[ℓ]n→[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 G can be done with O(n3/2) 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(n3/4m+n) in the adjacency list model.

Highlights

Query complexity of a function f : [ ]n → [m] is the minimum number of adaptive queries to its input bits required to compute the output of the function
The trivial classical algorithm for the search problem which queries the input bits one by one have query complexity T = n, and the guessing algorithm which always predicts the output 0 makes at most G = 1 mistakes
The optimization is over vectors |uxj, |wxj. This semi-definite program (SDP) is called the dual adversary bound and is proved by Lee et al [LMR+11] to be an upper bound on quantum query complexity of the function f as well

Summary

Introduction

Query complexity of a function f : [ ]n → [m] is the minimum number of adaptive queries to its input bits required to compute the output of the function. The trivial classical algorithm for the search problem which queries the input bits one by one have query complexity T = n, and the guessing algorithm which always predicts the output 0 makes at most G = 1 mistakes (because making a mistake is equivalent to finding an input bit 1 which solves the√search prob√lem). Inspired by this proof, we generalize Lin and Lin’s result for functions f : [ ]n → [m] with non-binary input as well as non-binary output alphabets Our proof of this generalization is based on the dual adversary bound which is another equivalent characterization of the quantum query complexity [LMR+11]. To the authors’ knowledge this is the first non-trivial upper bound for this problem

Preliminaries

Query algorithms

Dual adversary bound

Non-binary span program

From decision trees to span programs

Main result: generalization to the non-binary case

Wblack Eζ

Applications

Graph properties in the adjacency matrix model

9: Return the DFS tree

Graph properties in the adjacency list model

Concluding remarks