Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.

in the SQ model is often taken as strong evidence for hardness in the sample model.In summary, we study in this work the following problems, which are stated more formally in Section 3:Problems: PAC probabilistic modelling of quantum circuit Born machines (informal). Let C be the set of output distributions corresponding to a class of local quantum circuits. Given either sample-oracle or SQ-oracle access to some unknown distribution P ∈ C, output, with high probability, either generative modelling: an efficient generator, or density modelling: an efficient evaluator for a distribution P which is sufficiently close to P .If there exists either a sample or computationally efficient algorithm which, with respect to either the sample oracle or the SQ oracle, solves the generative (density) modelling problem associated with a given set of distributions C, then we say that C is sample or computationally efficiently generator (evaluator) learnable within the relevant oracle model. We are particularly interested in this work in establishing the existence or non-existence, of efficient quantum or classical learning algorithms, for the output distributions of various classes of local quantum circuits, within both the sample and statistical query model. Main resultsGiven this motivation and context, we provide two main results, which stated informally, are as follows:Result 1 (Informal version of Corollary 1). The concept class consisting of the output distributions of super-logarithmic depth nearest neighbour Clifford circuits is not sample efficiently PAC generator-learnable or evaluator-learnable, in the statistical query model.Result 2 (Informal version of Theorem 2). The concept class consisting of the output distributions of nearest neighbour Clifford circuits is both sample and computationally efficiently classically PAC generator-learnable and evaluator-learnable, in the sample model.

Strongly correlated quantum many-body systems at low dimension exhibit a wealth of phenomena, ranging from features of geometric frustration to signatures of symmetry-protected topological order. In suitable descriptions of such systems, it can be helpful to resort to effective models which focus on the essential degrees of freedom of the given model. In this work, we analyze how to determine the validity of an effective model by demanding it to be in the same phase as the original model. We focus our study on one-dimensional spin-1/2 systems and explain how non-trivial symmetry protected topologically ordered (SPT) phases of an effective spin 1 model can arise depending on the couplings in the original Hamiltonian. In this analysis, tensor network methods feature in two ways: On the one hand, we make use of recent techniques for the classification of SPT phases using matrix product states in order to identify the phases in the effective model with those in the underlying physical system, employing Künneth's theorem for cohomology. As an intuitive paradigmatic model we exemplify the developed methodology by investigating the bi-layered ∆-chain. For strong ferromagnetic inter-layer couplings, we find the system to transit into exactly the same phase as an effective spin 1 model. However, for weak but finite coupling strength, we identify a symmetry broken phase differing from this effective spin-1 description. On the other hand, we underpin our argument with a numerical analysis making use of matrix product states. arXiv:1704.02992v1 [cond-mat.str-el]

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