  The Schmidt decomposition is the go-to tool for measuring bipartite
entanglement of pure quantum states. Similarly, it is possible to study the
entangling features of a quantum operation using its operator-Schmidt, or
tensor product decomposition. While quantum technological implementations of
the former are thoroughly studied, entangling properties on the operator level
are harder to extract in the quantum computational framework because of the
exponential nature of sample complexity. Here we present an algorithm for
unbalanced partitions into a small subsystem and a large one (the environment)
to compute the tensor product decomposition of a unitary whose effect on the
small subsystem is captured in classical memory while the effect on the
environment is accessible as a quantum resource. This quantum algorithm may be
used to make predictions about operator non-locality, effective open quantum
dynamics on a subsystem, as well as for finding low-rank approximations and
low-depth compilations of quantum circuit unitaries. We demonstrate the method
and its applications on a time-evolution unitary of an isotropic Heisenberg
model in two dimensions.
