Separating Quantum and Classical Learning

Abstract

Abstract. We consider a model of learning Boolean functions from quantum membership queries. This model was studied in [26], where it was shown that any class of Boolean functions which is information-theoretically learnable from polynomially many quantum membership queries is also information-theoretically learnable from polynomially many classical membership queries. In this paper we establish a strong computational separation between quantum and classical learning. We prove that if any cryptographic one-way function exists, then there is a class of Boolean functions which is polynomial-time learnable from quantum membership queries but not polynomial-time learnable from classical membership queries. A novel consequence of our result is a quantum algorithm that breaks a general cryptographic construction which is secure in the classical setting.

Quantum speed-up for unsupervised learning

“…A quantum analogue of Angluin's model has been defined by Servedio (2001). The main difference is that our goal is to perform clustering and not to learn exactly a function computed by an oracle.…”

Quantum inductive inference by finite automata

“…In the next Section, we describe the variants of Gold's model that we then examine in a quantum setting. We note that quantum versions of other models of learning have been considered by other authors, in [30,31,21] for example, but of different nature than ours'.…”

Quantum clustering algorithms

“…This model is close in spirit to the one imagined by Angluin (1988), which is used in computational learning theory to study the query complexity of learning a function given by a black box. A quantum analogue of Angluin's model has been defined by Servedio (2001). The main difference between Angluin's model and ours is that we are not interested in learning a function but rather in performing clustering 4 .…”