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Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for representing a matrix from a set of selected rows and columns-is an efficient method for constructing a tensor train decomposition of a tensor from few of its entries. While tensor train cross approximation has achieved remarkable performance in practical applications, its theoretical analysis, in particular regarding the error of the approximation, is so far lacking. To our knowledge, existing results only provide element-wise approximation accuracy guarantees, which lead to a very loose bound when extended to the entire tensor. In this paper, we bridge this gap by providing accuracy guarantees in terms of the entire tensor for both exact and noisy measurements. Our results illustrate how the choice of selected subtensors affects the quality of the cross approximation and that the approximation error caused by model error and/or measurement error may not grow exponentially with the order of the tensor. These results are verified by numerical experiments, and may have important implications for the usefulness of cross approximations for high-order tensors, such as those encountered in the description of quantum many-body states. 

It has been recently shown that a state generated by a one-dimensional noisy quantum computer is well approximated by a matrix product operator with a finite bond dimension independent of the number of qubits. We show that full quantum state tomography can be performed for such a state with a minimal number of measurement settings using a method known as tensor train cross approximation. The method works for reconstructing full rank density matrices and only requires measuring local operators, which are routinely performed in state-of-art experimental quantum platforms. Our method requires exponentially fewer state copies than the best known tomography method for unstructured states and local measurements. The fidelity of our reconstructed state can be further improved via supervised machine learning, without demanding more experimental data. Scalable tomography is achieved if the full state can be reconstructed from local reductions.