  Quantum annealers manufactured by D-Wave Systems, Inc., are computational
devices capable of finding high-quality solutions of NP-hard problems. In this
contribution, we explore the potential and effectiveness of such quantum
annealers for computing Boolean tensor networks. Tensors offer a natural way to
model high-dimensional data commonplace in many scientific fields, and
representing a binary tensor as a Boolean tensor network is the task of
expressing a tensor containing categorical (i.e., {0, 1}) values as a product
of low dimensional binary tensors. A Boolean tensor network is computed by
Boolean tensor decomposition, and it is usually not exact. The aim of such
decomposition is to minimize the given distance measure between the
high-dimensional input tensor and the product of lower-dimensional (usually
three-dimensional) tensors and matrices representing the tensor network. In
this paper, we introduce and analyze three general algorithms for Boolean
tensor networks: Tucker, Tensor Train, and Hierarchical Tucker networks. The
computation of a Boolean tensor network is reduced to a sequence of Boolean
matrix factorizations, which we show can be expressed as a quadratic
unconstrained binary optimization problem suitable for solving on a quantum
annealer. By using a novel method we introduce called \textit{parallel quantum
annealing}, we demonstrate that tensor with up to millions of elements can be
decomposed efficiently using a DWave 2000Q quantum annealer.
