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 O(γ^Clog∣C∣),
where γ^C 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
O(γ^Clog∣C∣loglog∣C∣) 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 (ϵ,δ)-PAC
learning any concept class of Vapnik-Chervonenkis dimension d over the domain
{0,1}n from Ω(nd) to Ω(ϵ1logδ1+d+ϵd). 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
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