Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constantsized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the element distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph.We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit.We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity O(n 35/27 ) that is better than O(n 13/10 ) of the best previously known algorithm by Magniez et al.

We introduce a span program that decides st-connectivity, and generalize the span program to develop quantum algorithms for several graph problems. First, we give an algorithm for st-connectivity that uses O(n √ d) quantum queries to the n × n adjacency matrix to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also show that if T is a path, a star with two subdivided legs, or a subdivision of a claw, its presence as a subgraph in the input graph G can be detected with O(n) quantum queries to the adjacency matrix. Under the promise that G either contains T as a subgraph or does not contain T as a minor, we give O(n)-query quantum algorithms for detecting T either a triangle or a subdivision of a star. All these algorithms can be implemented time efficiently and, except for the triangle-detection algorithm, in logarithmic space. One of the main techniques is to modify the st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.

We present a quantum algorithm solving the k-distinctness problem in O n 1−2 k−2 /(2 k −1) queries with a bounded error. This improves the previous O(n k/(k+1) )-query algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee [7], the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler.Additionally, we introduce an O( √ nα 1/6 ) algorithm for the graph collision problem where α is the independence number of the graph. * Faculty of Computing, University of Latvia, stiboh@gmail.com. certificate complexity C x (f ) of f on x is defined as the minimal size of a certificate for f that x satisfies. The b-certificate complexity C (b) (f ) is defined as max x∈f −1 (b) C x (f ). Thus, for instance, 1-certificate complexity of element distinctness is 2, and 1-certificate complexity of triangle detection is 3.Soon after the Ambainis' paper, it was realized [14] that the algorithm developed for k-distinctness can be used to evaluate, in the same number of queries, any function with 1-certificate complexity equal to k. Now we know that for some functions this algorithm is tight, due to the lower bound for the k-sum problem [9]. The goal of the k-sum problem is to detect, given n elements of an Abelian group as input, whether there are k of them that sum up to a prescribed element of the group. The k-sum problem is noticeable in the sense that, given any (k − 1)-tuple of input elements, one has absolutely no information on whether they form a part of an (inclusion-wise minimal) 1-certificate, or not.The aforementioned applications of the quantum walk on the Johnson graph (triangle finding, etc.) went beyond O(n k/(k+1) ) upper bound by utilizing additional relations between the input variables: the adjacency relation of the edges for the triangle problem, row-column relations for the matrix products, and so on. For instance, two edges in a graph can't be a part of a 1-certificate for the triangle problem, if they are not adjacent.The k-distinctness problem is different in the sense that it doesn't possess any structure of the variables. But it does possess a relation between the values of the variables: two elements can't be a part of a 1-certificate if their values are different. However, it seems that quantum walk on the Johnson graph fails to utilize this structure efficiently.In this paper, we use the learning graph approach to construct a quantum algorithm that solves the k-distinctness problem in O n 1−2 k−2 /(2 k −1) queries. Note that O n 1−2 k−2 /(2 k −1) = o(n 3/4 ). Thus, our algorithm solves k-distinctness, for arbitrary k, in asymptotically less queries than the best previously known algorithm solves 3-distinctness.The learning graph is a novel way of construction quantum query algorithms. Somehow, it may be thought as a way of designing a more flexible quantum walk than just on the Johnson graph. And compared to the quantum walk design paradigms from Ref. [25,22], it is easier to deal with. ...

We prove a tight quantum query lower bound Ω(n k/(k+1) ) for the problem of deciding whether there exist k numbers among n that sum up to a prescribed number, provided that the alphabet size is sufficiently large.

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein.Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that nonadaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it.For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the k-sum problem derived recently [1].Finally, we use these results to show that the learning graph for the triangle problem from [2] is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.