  Finding ways to transform a quantum state to another is fundamental to
quantum information processing. In this paper, we apply the sparse matrix
approach to the quantum state transformation problem. In particular, we present
a new approach for searching for unitary matrices for quantum state
transformation by directly optimizing the objective problem using the
Alternating Direction Method of Multipliers (ADMM). Moreover, we consider the
use of group sparsity as an alternative sparsity choice in quantum state
transformation problems. Our approach incorporates sparsity constraints into
quantum state transformation by formulating it as a non-convex problem. It
establishes a useful framework for efficiently handling complex quantum systems
and achieving precise state transformations.
