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The constantly increasing dimensionality of artificial quantum systems demands for highly efficient methods for their characterization and benchmarking. Conventional quantum tomography fails for larger systems due to the exponential growth of the required number of measurements. The conceptual solution for this dimensionality curse relies on a simple idea - a complete description of a quantum state is excessive and can be discarded in favor of experimentally accessible information about the system. The probably approximately correct (PAC) learning theory has been recently successfully applied to a problem of building accurate predictors for the measurement outcomes using a dataset which scales only linearly with the number of qubits. Here we present a constructive and numerically efficient protocol which learns a tensor network model of an unknown quantum system. We discuss the limitations and the scalability of the proposed method., Comment: 10 pages, 11 figures

We develop the learning algorithm to build the architecture agnostic model of the reconfigurable optical interferometer. Programming the unitary transformation on the optical modes of the interferometer either follows the analytical expression yielding the unitary matrix given the set of phaseshifts or requires the optimization routine if the analytic decomposition does not exist. Our algorithm adopts the supervised learning strategy which matches the model of the interferometer to the training set populated by the samples produced by the device under study. The simple optimization routine uses the trained model to output the phaseshifts of the interferometer with the given architecture corresponding to the desired unitary transformation. Our result provides the recipe for efficient tuning of the interferometers even without rigorous analytical description which opens opportunity to explore new architectures of the interferometric circuits., Comment: 10 pages, 6 figures

Multiple methods for deriving the energy-momentum tensor for a physical theory exist in the literature. The most common methods are to use Noether's first theorem with the 4-parameter Poincar\'{e} translation, or to write the action in a curved spacetime and perform variation with respect to the metric tensor, then return to a Minkowski spacetime. These are referred to as the Noether and Hilbert (metric/ curved space/ variational) energy-momentum tensors, respectively. In electrodynamics and other simple models, the Noether and Hilbert methods yield the same result. Due to this fact, it is often asserted that these methods are generally equivalent for any theory considered, and that this gives physicists a freedom in using either method to derive an energy-momentum tensor depending on the problem at hand. $\dots$ For spin-2, the ideal candidate to check this equivalence for a more complicated model, there exist many energy-momentum tensors in the literature, none of which are gauge invariant, so it is not clear which expression one hopes to obtain from the Noether and Hilbert approaches unlike in the case of e.g. electrodynamics. It has been shown, however, that the linearized Gauss-Bonnet gravity model (second order derivatives, second rank tensor potential) has an energy-momentum tensor that is unique, gauge invariant, symmetric, conserved, and trace-free when derived from Noether's first theorem (all the same properties of the physical energy-momentum tensor of electrodynamics). This makes it the ideal candidate to check if the Noether and Hilbert methods coincide for a more complicated model. It is proven here using this model as a counterexample, by direct calculation, that the Noether and Hilbert energy-momentum tensors are not, in general, equivalent., Comment: 35 pages

A connection between linearized Gauss-Bonnet gravity and classical electrodynamics is found by developing a procedure which can be used to derive completely gauge invariant models. The procedure involves building the most general Lagrangian for a particular order of derivatives ($N$) and rank of tensor potential ($M$), then solving such that the model is completely gauge invariant (the Lagrangian density, equation of motion and energy-momentum tensor are all gauge invariant). In the case of $N = 1$ order of derivatives and $M = 1$ rank of tensor potential, electrodynamics is uniquely derived from the procedure. In the case of $N = 2$ order of derivatives and $M = 2$ rank of symmetric tensor potential, linearized Gauss-Bonnet gravity is uniquely derived from the procedure. The natural outcome of the models for classical electrodynamics and linearized Gauss-Bonnet gravity from a common set of rules provides an interesting connection between two well explored physical models., Comment: 12 pages