  The agnostic setting is the hardest generalization of the PAC model since it
is akin to learning with adversarial noise. In this paper, we give a
poly$(n,t,{\frac{1}{\varepsilon}})$ quantum algorithm for learning size $t$
decision trees with uniform marginal over instances, in the agnostic setting,
without membership queries. Our algorithm is the first algorithm (classical or
quantum) for learning decision trees in polynomial time without membership
queries. We show how to construct a quantum agnostic weak learner by designing
a quantum version of the classical Goldreich-Levin algorithm that works with
strongly biased function oracles. We show how to quantize the agnostic boosting
algorithm by Kalai and Kanade (NIPS 2009) to obtain the first efficient quantum
agnostic boosting algorithm. Our quantum boosting algorithm has a polynomial
improvement in the dependence of the bias of the weak learner over all adaptive
quantum boosting algorithms while retaining the standard speedup in the VC
dimension over classical boosting algorithms. We then use our quantum boosting
algorithm to boost the weak quantum learner we obtained in the previous step to
obtain a quantum agnostic learner for decision trees. Using the above
framework, we also give quantum decision tree learning algorithms for both the
realizable setting and random classification noise model, again without
membership queries.
