Quantum Examples

Dick H. Holze

Nader H. Bshouty, Jeffrey C. Jackson

We generalize the notion of probably approximately correct (PAC) learning from an example oracle to a notion of efficient learning on a quantum computer using a quantum example oracle. This quantum example oracle is a natural extension of the traditional PAC example oracle, and it immediately follows that all PAC-learnable function classes are learnable in the quantum model. Furthermore, we obtain positive quantum learning results for classes that are not known to be PAC learnable. Specifically, we show that disjunctive normal form (DNF) is efficiently learnable with respect to the uniform distribution by a quantum algorithm using a quantum example oracle. While it was already known that DNF is uniform-learnable using a membership oracle, we prove that a quantum example oracle with respect to uniform is less powerful than a membership oracle.

Diederik Aerts, Thomas Durt, Bruno Van Bogaert

We present an explicit physical example of an experimental situation on a physical entity that gives rise to a fuzzy set. The fuzziness in the example is due to ∞uctuations of the experimental apparatus, and not to an indeterminacy about the states of the physical entity, and is described by a varying parameter †. For zero value of the parameter (no ∞uctuations), the example reduces to a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal value (maximal ∞uctuations), the example gives rise to a quantum fuzzy set, more precisely a spin- model. In between, we have a continuum of fuzzy situations, neither classical, nor quantum. We believe that the example can make us understand the nature of the quantum mechanical fuzziness and probability, and how these are related to the classical situation.

Arif Ahmed

Jinge Yan, Lili Yan, Shibin Zhang

A quantum variational circuit is a quantum machine learning model similar to a neural network. A crafted adversarial example can lead to incorrect results for the model. Using adversarial examples to train the model will greatly improve its robustness. The existing method is to use automatic differentials or finite difference to obtain a gradient and use it to construct adversarial examples. This paper proposes an innovative method for constructing adversarial examples of quantum variational circuits. In this method, the gradient can be obtained by measuring the expected value of a quantum bit respectively in a series quantum circuit. This method can be used to construct the adversarial examples for a quantum variational circuit classifier. The implementation results prove the effectiveness of the proposed method. Compared with the existing method, our method requires fewer resources and is more efficient.