Learning with Quantum Models

Srinivasan Arunachalam, Alex B. Grilo, Henry Yuen

We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.

Søren Gammelmark, Klaus Mølmer

Kyungmin Lee, Hyungjun Jeon, Dongkyu Lee +3 more

Abstract Variational quantum machine learning (VQML) models based on parameterized quantum circuits (PQC) have been expected to offer a potential quantum advantage for machine learning (ML) applications. However, comparison between VQML models and their classical counterparts is hard due to the lack of interpretability of VQML models. In this study, we introduce a graphical approach to analyze the PQC and the corresponding operation of VQML models to deal with this problem. In particular, we utilize the Stokes representation of quantum states to treat VQML models as network models based on the corresponding representations of basic gates. From this approach, we suggest the notion of active paths in the networks and relate the expressivity of VQML models with it. We investigate the growth of active paths in VQML models and observe that the expressivity of VQML models can be significantly limited for certain cases. Then we construct classical models inspired by our graphical interpretation of VQML models and show that they can emulate or outperform the outputs of VQML models for these cases. Our result provides a new way to interpret the operation of VQML models and facilitates the interconnection between quantum and classical ML areas.

Mike Hernacki

GÜLAY ERDEM, NALAN BALKIR

Takao Tomono, Satoko Natsubori

Abstract Quantum machine learning is one of most exciting prospective applications of quantum technologies. Quantum kernel SVM is one of most popular methods for classifiers. However, there are few reports that show the quantum advantage in learning model construction compared with classical learning models. In this work, we investigated the process of learning model construction of quantum machine learning by using an evaluation index and ROC space graph. The behaviour between quantum and classical machine learning is different, and the quantum learning model is the method to decrease the false positive rate from high false and true positive rates that is based on quantum superposition. In summary, we have demonstrated the effect of quantum superposition on quantum machine learning for the first time from actual experimental results. This truth supports that quantum machine learning can build an outline of learning models faster than classical methods even if the number of original data is smaller.