  Discrete-time quantum walks with position-dependent coin operators have
numerous applications. For a position dependence that is sufficiently smooth,
it has been provided in Ref. [1] an approximate quantum-circuit implementation
of the coin operator that is efficient. If we want the quantum-circuit
implementation to be exact (e.g., either, in the case of a smooth position
dependence, to have a perfect precision, or in order to treat a non-smooth
position dependence), but the depth of the circuit not to scale exponentially,
then we can use the linear-depth circuit of Ref. [1], which achieves a depth
that is linear at the cost of introducing an exponential number of ancillas. In
this paper, we provide an adjustable-depth quantum circuit for the exact
implementation of the position-dependent coin operator. This adjustable-depth
circuit consists in (i) applying in parallel, with a linear-depth circuit, only
certain packs of coin operators (rather than all of them as in the original
linear-depth circuit [1]), each pack contributing linearly to the depth, and in
(ii) applying sequentially these packs, which contributes exponentially to the
depth.
