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.
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.
Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002], 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 Q(f)=Θ(B(f)). 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 O(n1.5logn) [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite matching problem gives an O(n1.75) algorithm, improving the best known trivial O(n2) upper bound.
We describe a method to upper bound the quantum query complexity of Boolean formula evaluation problems, using fundamental theorems about the general adversary bound.This nonconstructive method can give an upper bound on query complexity without producing an algorithm.For example, we describe an oracle problem which we prove (nonconstructively) can be solved in O(1) queries, where the previous best quantum algorithm uses a polylogarithmic number of queries.We then give an explicit O(1)-query algorithm for this problem based on span programs.
In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.
We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2) bound for finding a maximum-out-degree vertex in an n-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an n-vertex undirected graph requires Θ(n3/2) quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum (k,n−k)-Cut, which asks for the largest cut with one side of size k. For every constant k≥1, we give an O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)-Cut involving k-Graph Collision, which asks whether the marked vertices of a known graph contain a k-clique. For k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for k-Graph Collision, the reduction yields a lower bound matching our upper bound above.
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.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most q oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model:
(i) (Unbiased) Phase estimation up to a precision ε requires Θ(qε21) queries;
(ii) (Unbiased) Amplitude estimation up to a precision ε requires Θ(qε2p(1−p)) queries;
(iii) Search among N items requires Θ(qN) queries;
(iv) Two level AND-OR tree (AND of m ORs, with n inputs to each OR) requires Θ(qnm) queries.
All of these bounds are optimal up to a constant factor, for all q from 1, corresponding to the classical complexity, to Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor.
Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.