  Realizing controlled operations is fundamental to the design and execution of
quantum algorithms. In quantum simulation and learning of quantum many-body
systems, an important subroutine consists of implementing a controlled
Hamiltonian time-evolution. Given only black-box access to the uncontrolled
evolution $e^{-iHt}$, controlizing it, i.e., implementing
$\mathrm{ctrl}(e^{-iHt}) = |0\rangle\langle 0|\otimes I + |1\rangle\langle 1
|\otimes e^{-iHt}$ is non-trivial. A recent paper introduced a controlization
scheme in the context of designing algorithms for transforming unknown
Hamiltonian dynamics [OKTM24]. The main idea behind the scheme is to
intersperse the uncontrolled evolution with suitable operations such that the
overall dynamics approximates the desired controlled evolution. The scheme in
Ref. [OKTM24], although efficient, uses operations randomly sampled from an
exponentially large set. In the present work, we show that more efficient
controlization schemes can be constructed with the help of orthogonal arrays
for unknown 2-local Hamiltonians. This construction can also be generalized to
$k$-local Hamiltonians. Moreover, our controlization schemes based on
orthogonal arrays can take advantage of the interaction graph's structure and
be made more efficient.
