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. [17] 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 at most O( log |C| √ γ̂ ), where γ̂ 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 at most O( log |C| log log |C| √ γ̂ ) 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 (ǫ, δ)-PAC learning any concept class of VapnikChervonenkis dimension d over the domain {0, 1}n from Ω( d n ) to Ω( ǫ ln 1 δ + d + √ d ǫ ). This new lower bound comes closer to matching known upper bounds for classical PAC learning.