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arXiv:1905.13095 · Quantum

Quantum Speedup Based on Classical Decision Trees

Salman Beigi, Leila Taghavi

Published Mar 2, 2020• 17 citations
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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.

DOI
10.22331/q-2020-03-02-241

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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.
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s .Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree.Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.
It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity O( √ GT where T is the query complexity of the classical algorithm (depth of the decision tree) and G is the maximum number of wrong answers by the guessing algorithm [ 3 , 14 ]. In this article, we show that, given some constraints on the classical algorithms, this quantum algorithm can be implemented in time Õ (√ GT ). Our algorithm is based on non-binary span programs and their efficient implementation. We conclude that various graph-theoretic problems including bipartiteness, cycle detection, and topological sort can be solved in time O(n 3/2 log 2 n ) and with O(n 3/2 ) quantum queries. Moreover, finding a maximal matching can be solved with O(n 3/2 ) quantum queries in time O(n 3/2 log 2 n ), and maximum bipartite matching can be solved in time O(n 2 log 2 n ).
We consider the following question in query complexity: Given a classical query algorithm in the form of a decision tree, when does there exist a quantum query algorithm with a speed-up (i.e., that makes fewer queries) over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up in the number of queries. In particular, our results give a bounded-error quantum query algorithm of cost O(√s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s. This recovers an O(√n) algorithm for the Search problem, for example. Lin and Lin [Theory of Computing'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor. Their upper bounds are in terms of a quantity which we call the “guessing complexity” of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Information and Computation'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any measure studied in classical complexity theory. We also show a polynomial separation between rank and its natural randomized analog for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to edges of the tree. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are strictly stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes in their construction could yield better upper bounds.
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s)O(\sqrt{s})O(s​) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size sss. Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.
Span programs characterize the quantum query complexity of binary functions f:\{0,\ldots,\ell\}^n \to \{0,1\} up to a constant factor. In this paper we generalize the notion of span programs for functions with non-binary input/output alphabets f: [\ell]^n \to [m]. We show that non-binary span program characterizes the quantum query complexity of any such function up to a constant factor. We argue that this non-binary span program is indeed the generalization of its binary counterpart. We also generalize the notion of span programs for a special class of relations. Learning graphs provide another tool for designing quantum query algorithms for binary functions. In this paper, we also generalize this tool for non-binary functions, and as an application of our non-binary span program show that any non-binary learning graph gives an upper bound on the quantum query complexity.
Inspired by the Elitzur-Vaidman bomb testing problem [19], we introduce a new query complexity model, which we call bomb query complexity B(f). We investigate its relationship with the usual quantum query complexity Q(f), and show that B(f) = Θ(Q(f)2).This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on [EQUATION]. We subsequently were able to give explicit quantum algorithms matching our upper bound method. We apply this method on the single-source shortest paths problem on unweighted graphs, obtaining an algorithm with O(n1.5) quantum query complexity, improving the best known algorithm of [EQUATION] [21]. Applying this method to the maximum bipartite matching problem gives an O(n1.75) algorithm, improving the best known trivial O(n2) upper bound.