On the Power of Non-Adaptive Learning Graphs

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 non-adaptive 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 in arXiv:1206.6528. Finally, we use these results to show that the learning graph for the triangle problem from arXiv:1210.1014 is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.Comment: 16 pages, 1.5 figures v2: the main result generalised for all certificate structures, a bug in the proof of Proposition 17 fixe

Adversary Lower Bound for Element Distinctness with Small Range

The Element Distinctness problem is to decide whether each character of an input string is unique. The quantum query complexity of Element Distinctness is known to be $\Theta(N^{2/3})$; the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size $\Omega(N^2)$. We construct a tight $\Omega(N^{2/3})$ adversary lower bound for Element Distinctness with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.Comment: 22 pages. v2: one figure added, updated references, and minor typos fixed. v3: minor typos fixe

An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.Comment: 12 page

Quantum Algorithms for Computational Geometry Problems

We study quantum algorithms for problems in computational geometry, such as POINT-ON-3-LINES problem. In this problem, we are given a set of lines and we are asked to find a point that lies on at least $3$ of these lines. POINT-ON-3-LINES and many other computational geometry problems are known to be 3SUM-HARD. That is, solving them classically requires time $\Omega(n^{2-o(1)})$, unless there is faster algorithm for the well known 3SUM problem (in which we are given a set $S$ of $n$ integers and have to determine if there are $a, b, c \in S$ such that $a + b + c = 0$). Quantumly, 3SUM can be solved in time $O(n \log n)$ using Grover's quantum search algorithm. This leads to a question: can we solve POINT-ON-3-LINES and other 3SUM-HARD problems in $O(n^c)$ time quantumly, for $c<2$? We answer this question affirmatively, by constructing a quantum algorithm that solves POINT-ON-3-LINES in time $O(n^{1 + o(1)})$. The algorithm combines recursive use of amplitude amplification with geometrical ideas. We show that the same ideas give $O(n^{1 + o(1)})$ time algorithm for many 3SUM-HARD geometrical problems.Comment: 10 page

Adversary Lower Bound for the Orthogonal Array Problem

We prove a quantum query lower bound \Omega(n^{(d+1)/(d+2)}) for the problem of deciding whether an input string of size n contains a k-tuple which belongs to a fixed orthogonal array on k factors of strength d<=k-1 and index 1, provided that the alphabet size is sufficiently large. Our lower bound is tight when d=k-1. The orthogonal array problem includes the following problems as special cases: k-sum problem with d=k-1, k-distinctness problem with d=1, k-pattern problem with d=0, (d-1)-degree problem with 1<=d<=k-1, unordered search with d=0 and k=1, and graph collision with d=0 and k=2.Comment: 13 page

Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks

Many computational problems are subject to a quantum speed-up: one might find that a problem having an Opn3q-time or Opn2q-time classic algorithm can be solved by a known Opn1.5q-time or Opnq-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro and Speelman [7], and by Aaronson, Chia, Lin, Wang, and Zhang [1]. In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time Opn2´εq, for any ε ą 0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear Opn1´εq-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems. These results are proven by adapting to the quantum setting known classical fine-grained reductions from the 3SUM problem. This adaptation is not trivial, however, since the original classical reductions require pre-processing the input in various ways, e.g. by sorting it according to some order, and this pre-processing (provably) cannot be done in sublinear quantum time. We overcome this bottleneck by combining a quantum walk with a classical dynamic data-structure having a certain “history-independence” property. This type of construction has been used in the past to prove upper bounds, and here we use it for the first time as part of a reduction. This general proof strategy allows us to prove tight lower bounds on several computational-geometry problems, on Convolution-3SUM and on the 0-Edge-Weight-Triangle problem, conditional on the Quantum-3SUM-Conjecture. We believe this proof strategy will be useful in proving tight (conditional) lower-bounds, and limits on quantum speed-ups, for many other problems