48,544 research outputs found
Improved Bounds on Quantum Learning Algorithms
In this article we give several new results on the complexity of algorithms
that learn Boolean functions from quantum queries and quantum examples.
Hunziker et al. conjectured that for any class C of Boolean functions, the
number of quantum black-box queries which are required to exactly identify an
unknown function from C is ,
where is a combinatorial parameter of the class C. We
essentially resolve this conjecture in the affirmative by giving a quantum
algorithm that, for any class C, identifies any unknown function from C using
quantum black-box
queries.
We consider a range of natural problems intermediate between the exact
learning problem (in which the learner must obtain all bits of information
about the black-box function) and the usual problem of computing a predicate
(in which the learner must obtain only one bit of information about the
black-box function). We give positive and negative results on when the quantum
and classical query complexities of these intermediate problems are
polynomially related to each other.
Finally, we improve the known lower bounds on the number of quantum examples
(as opposed to quantum black-box queries) required for -PAC
learning any concept class of Vapnik-Chervonenkis dimension d over the domain
from to . This new lower bound comes
closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information
Processing. Requires: algorithm.sty, algorithmic.sty to buil
Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems
We deal with the problem, initiated in [8], of finding randomized and quantum
complexity of initial-value problems. We showed in [8] that a speed-up in both
settings over the worst-case deterministic complexity is possible. In the
present paper we prove, by defining new algorithms, that further improvement in
upper bounds on the randomized and quantum complexity can be achieved. In the
H\"older class of right-hand side functions with r continuous bounded partial
derivatives, with r-th derivative being a H\"older function with exponent \rho,
the \epsilon-complexity is shown to be O((1/\epsilon)^{1/(r+\rho+1/3)}) in the
randomized setting, and O((1/\epsilon)^{1/(r+\rho+1/2)}) on a quantum computer
(up to logarithmic factors). This is an improvement for the general problem
over the results from [8]. The gap still remaining between upper and lower
bounds on the complexity is further discussed for a special problem. We
consider scalar autonomous problems, with the aim of computing the solution at
the end point of the interval of integration. For this problem, we fill up the
gap by establishing (essentially) matching upper and lower complexity bounds.
We show that the complexity in this case is of order
(1/\epsilon)^{1/(r+\rho+1/2)} in the randomized setting, and
(1/\epsilon)^{1/(r+\rho+1)} in the quantum setting (again up to logarithmic
factors).Comment: 17 pages, extended version (new section added), to appear in the
Journal of Complexit
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