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

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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

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