39 research outputs found
Quantum Algorithms for Classical Probability Distributions
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships.
Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions.
The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings
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
Adaptive Lower Bound for Testing Monotonicity on the Line
In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of epsilon-testing monotonicity of a function f : [n]->[r]. All our lower bounds are for adaptive two-sided testers.
- We prove a nearly tight lower bound for this problem in terms of r. The bound is Omega((log r)/(log log r)) when epsilon = 1/2. No previous satisfactory lower bound in terms of r was known.
- We completely characterise query complexity of this problem in terms of n for smaller values of epsilon. The complexity is Theta(epsilon^{-1} log (epsilon n)). Apart from giving the lower bound, this improves on the best known upper bound.
Finally, we give an alternative proof of the Omega(epsilon^{-1}d log n - epsilon^{-1}log epsilon^{-1}) lower bound for testing monotonicity on the hypergrid [n]^d due to Chakrabarty and Seshadhri (RANDOM\u2713)