  We report a deterministic and exact protocol to reverse any unknown
qubit-unitary operation, which simulates the time inversion of a closed qubit
system. To avoid known no-go results on universal deterministic exact unitary
inversion, we consider the most general class of protocols transforming unknown
unitary operations within the quantum circuit model, where the input unitary
operation is called multiple times in sequence and fixed quantum circuits are
inserted between the calls. In the proposed protocol, the input qubit-unitary
operation is called 4 times to achieve the inverse operation, and the output
state in an auxiliary system can be reused as a catalyst state in another run
of the unitary inversion. We also present the simplification of the
semidefinite programming for searching the optimal deterministic unitary
inversion protocol for an arbitrary dimension presented by M. T. Quintino and
D. Ebler [Quantum $\textbf{6}$, 679 (2022)]. We show a method to reduce the
large search space representing all possible protocols, which provides a useful
tool for analyzing higher-order quantum transformations for unitary operations.
