  Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in
mathematics, has been proposed originally for describing an (especially
one-dimensional) quantum system, and recently has found applications in various
applications such as compressing high-dimensional data, supervised kernel
linear classifier, and unsupervised generative modeling. However, when applied
to systems which are not defined on one-dimensional lattices, a serious
drawback of the MPS is the exponential decay of the correlations, which limits
its power in capturing long-range dependences among variables in the system. To
alleviate this problem, we propose to introduce long-range interactions, which
act as shortcuts, to MPS, resulting in a new model \textit{ Shortcut Matrix
Product States} (SMPS). When chosen properly, the shortcuts can decrease
significantly the correlation length of the MPS, while preserving the
computational efficiency. We develop efficient training methods of SMPS for
various tasks, establish some of their mathematical properties, and show how to
find a good location to add shortcuts. Finally, using extensive numerical
experiments we evaluate its performance in a variety of applications, including
function fitting, partition function calculation of $2-$d Ising model, and
unsupervised generative modeling of handwritten digits, to illustrate its
advantages over vanilla matrix product states.
