  The cost of data input can dominate the run-time of quantum algorithms. Here,
we consider data input of arithmetically structured matrices via block encoding
circuits, the input model for the quantum singular value transform and related
algorithms. We demonstrate how to construct block encoding circuits based on an
arithmetic description of the sparsity and pattern of repeated values of a
matrix. We present schemes yielding different subnormalisations of the block
encoding; a comparison shows that the best choice depends on the specific
matrix. The resulting circuits reduce flag qubit number according to sparsity,
and data loading cost according to repeated values, leading to an exponential
improvement for certain matrices. We give examples of applying our block
encoding schemes to a few families of matrices, including Toeplitz and
tridiagonal matrices.
