  We introduce the Quantum Unitary Reversal Algorithm (QURA), a deterministic
and exact approach to universally reverse arbitrary unknown unitary
transformations using $\mathcal{O}(d^2)$ calls of the unitary, where $d$ is the
system dimension. Our construction resolves a fundamental problem of
time-reversal simulations for closed quantum systems by affirming the
feasibility of reversing any unitary evolution without knowing the exact
process. The algorithm also provides the construction of a key oracle for
unitary inversion in quantum algorithm frameworks such as quantum singular
value transformation. Notably, our work demonstrates that compared with
classical methods relying on process tomography, reversing an unknown unitary
on a quantum computer holds a quadratic quantum advantage in computation
complexity. QURA ensures an exact unitary inversion while the classical
counterpart can never achieve exact inversion using a finite number of unitary
calls.
