466 research outputs found
Quantum Query Complexity of Subgraph Containment with Constant-sized Certificates
We study the quantum query complexity of constant-sized subgraph containment.
Such problems include determining whether an -vertex graph contains a
triangle, clique or star of some size. For a general subgraph with
vertices, we show that containment can be solved with quantum query
complexity , with a strictly positive
function of . This is better than \tilde{O}\s{n^{2-2/k}} by Magniez et
al. These results are obtained in the learning graph model of Belovs.Comment: 14 pages, 1 figure, published under title:"Quantum Query Complexity
of Constant-sized Subgraph Containment
Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
In this paper we present a quantum algorithm solving the triangle finding
problem in unweighted graphs with query complexity , where
denotes the number of vertices in the graph. This improves the previous
upper bound recently obtained by Lee, Magniez and
Santha. Our result shows, for the first time, that in the quantum query
complexity setting unweighted triangle finding is easier than its edge-weighted
version, since for finding an edge-weighted triangle Belovs and Rosmanis proved
that any quantum algorithm requires queries.
Our result also illustrates some limitations of the non-adaptive learning graph
approach used to obtain the previous upper bound since, even over
unweighted graphs, any quantum algorithm for triangle finding obtained using
this approach requires queries as well. To
bypass the obstacles characterized by these lower bounds, our quantum algorithm
uses combinatorial ideas exploiting the graph-theoretic properties of triangle
finding, which cannot be used when considering edge-weighted graphs or the
non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction
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 -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
Quantum Algorithms for Finding Constant-sized Sub-hypergraphs
We develop a general framework to construct quantum algorithms that detect if
a -uniform hypergraph given as input contains a sub-hypergraph isomorphic to
a prespecified constant-sized hypergraph. This framework is based on the
concept of nested quantum walks recently proposed by Jeffery, Kothari and
Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and
Santha [SODA'13] for similar problems over graphs. As applications, we obtain a
quantum algorithm for finding a -clique in a -uniform hypergraph on
vertices with query complexity , and a quantum algorithm for
determining if a ternary operator over a set of size is associative with
query complexity .Comment: 18 pages; v2: changed title, added more backgrounds to the
introduction, added another applicatio
The quantum complexity of approximating the frequency moments
The 'th frequency moment of a sequence of integers is defined as , where is the number of times that occurs in the
sequence. Here we study the quantum complexity of approximately computing the
frequency moments in two settings. In the query complexity setting, we wish to
minimise the number of queries to the input used to approximate up to
relative error . We give quantum algorithms which outperform the best
possible classical algorithms up to quadratically. In the multiple-pass
streaming setting, we see the elements of the input one at a time, and seek to
minimise the amount of storage space, or passes over the data, used to
approximate . We describe quantum algorithms for , and
in this model which substantially outperform the best possible
classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio
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