Quantum tomography by regularized linear regressions Abstract   In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or overcomplete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the traceone condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that      l    2     regularization with proper regularization gain provides even a lower meansquare error under a cost in bias. The optimal regularization parameter is defined in terms of a risk characterization for any finite sample size and a resulting implementable estimator is proposed. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed implementable tuning estimator is asymptotically optimal as the number of copies grows to infinity. Additionally, several numerical examples are provided to validate the established results.

Supervised learning with a quantum classifier using multilevel systems We propose a quantum classifier, which can classify data under the supervised learning scheme using a quantum feature space. The input feature vectors are encoded in a single quNit (a Nlevel quantum system), as opposed to more commonly used entangled multiqubit systems. For training, we use the much used quantum variational algorithmxe2x80x94a hybrid quantumxe2x80x93classical algorithm, in which the forward part of the computation is performed on a quantum hardware, whereas the feedback part is carried out on a classical computer. We introduce xe2x80x9csingleshot trainingxe2x80x9d, where all input samples belonging to the same class are used to train the classifier simultaneously. This significantly speeds up the training procedure and provides an advantage over classical machine learning classifiers. We demonstrate successful classification of popular benchmark datasets with our quantum classifier and compare its performance with respect to classical machine learning classifiers. We also show that the number of training parameters in our classifier is significantly less than the classical classifiers.