Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling. We also generalize previous results on full quantum state tomography in the online setting to (i) the $\epsilon$-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.

A non-linear activation function is an integral component of neural network algorithms used for various tasks such as data classification and pattern recognition. In neu-romorphic/emerging-hardware-based implementations of neural network algorithms, the non-linear activation function is often implemented through dedicated analog electronics. This enables faster execution of the activation function during training and inference of neural networks compared to conventional digital implementation. Here, with a similar motivation, we propose a novel non-linear activation function that can be used in a neural network for data classification. Our activation function can be implemented by taking advantage of the inherent nonlinearity in qubit preparation and SU(2) operation in quantum mechanics. These operations are directly implementable on quantum hardware through single-qubit quantum gates as we show here. In addition, the SU(2) parameters are adjustable here making the activation function adaptable; we adjust the parameters through classical feedback like in a variational algorithm in quantum machine learning. Using our proposed quantum activation function, we report accurate classification using popular machine learning data sets like Fisher's Iris, Wisconsin's Breast Cancer (WBC), Abalone, and MNIST on three different platforms: simulations on a classical computer, simulations on a quantum simulation framework like Qiskit, and experimental implementation on quantum hardware (IBM-Q). Then we use a Bloch-sphere-based approach to intuitively explain how our proposed quantum activation function, with its adaptability, helps in data classification.