Concurrently, we introduce the associated optimisation algorithm, which is inspired by GECO [25]--the latter does not always lead to good encodings (e.g., Sec.

We thank the reviewers for their invested time and constructive criticism. In the following, the comments are addressed separately for each reviewer. (line 58). We will try to expand this part of the paper. Hence, we obtain unrealistic interpolations similar to Figure 1 (bottom).

Quantum state tomography (QST) is the process of reconstructing the complete state of a quantum system (mathematically described as a density matrix) through a series of different measurements. These measurements are performed on a number of identical copies of the quantum system, with outcomes gathered as frequencies. QST aims to recover the density matrix and the corresponding properties of the quantum state from the measured frequencies. Although an informationally complete set of measurements can specify quantum state accurately in an ideal scenario with a large number of identical copies, both measurements and identical copies are restricted and imperfect in practical scenarios, making QST highly ill-posed. The conventional QST methods usually assume adequate or accurate measured frequencies or rely on manually designed regularizers to handle the ill-posed reconstruction problem, suffering from limited applications in realistic scenarios. Recent advances in deep neural networks (DNNs) led to the emergence of deep learning (DL) in QST. However, existing DL-based QST approaches often employ generic DNN models that are not optimized for imperfect conditions of QST. In this paper, we propose a transformer-based autoencoder architecture tailored for QST with imperfect measurement data. Our method leverages a transformer-based encoder to extract an informative latent representation (ILR) from imperfect measurement data and employs a decoder to predict the quantum states based on the ILR. We anticipate that the high-dimensional ILR will capture more comprehensive information about quantum states. To achieve this, we conduct pre-training of the encoder using a pretext task that involves reconstructing high-quality frequencies from measured frequencies. Extensive simulations and experiments demonstrate the remarkable ability of the ILR in dealing with imperfect measurement data in QST.

We address the problem of one-to-many mappings in supervised learning, where a single instance has many different solutions of possibly equal cost. The framework of conditional variational autoencoders describes a class of methods to tackle such structured-prediction tasks by means of latent variables. We propose to incentivise informative latent representations for increasing the generalisation capacity of conditional variational autoencoders. To this end, we modify the latent variable model by defining the likelihood as a function of the latent variable only and introduce an expressive multimodal prior to enable the model for capturing semantically meaningful features of the data. To validate our approach, we train our model on the Cornell Robot Grasping dataset, and modified versions of MNIST and Fashion-MNIST obtaining results that show a significantly higher generalisation capability.

We propose to learn a hierarchical prior in the context of variational autoencoders to avoid the over-regularisation resulting from a standard normal prior distribution. To incentivise an informative latent representation of the data by learning a rich hierarchical prior, we formulate the objective function as the Lagrangian of a constrained-optimisation problem and propose an optimisation algorithm inspired by Taming VAEs. We introduce a graph-based interpolation method, which shows that the topology of the learned latent representation corresponds to the topology of the data manifold---and present several examples, where desired properties of latent representation such as smoothness and simple explanatory factors are learned by the prior. Furthermore, we validate our approach on standard datasets, obtaining state-of-the-art test log-likelihoods.