  Quantum mechanics features a variety of distinct properties such as coherence
and entanglement, which could be explored to showcase potential advantages over
classical counterparts in information processing. In general, legitimate
quantum operations must adhere to principles of quantum mechanics, particularly
the requirements of complete positivity and trace preservation. Nonetheless,
non-physical maps, especially Hermitian-preserving maps, play a crucial role in
quantum information science. To date, there exists no effective method for
implementing these non-physical maps with quantum devices. In this work, we
introduce the Hermitian-preserving map exponentiation algorithm, which can
effectively realize the action of an arbitrary Hermitian-preserving map by
encoding its output into a quantum process. We analyze the performances of this
algorithm, including its sample complexity and robustness, and prove its
optimality in certain cases. When combined with algorithms such as the Hadamard
test and quantum phase estimation, it allows for the extraction of information
and generation of states from outputs of Hermitian-preserving maps, enabling
various applications. Utilizing positive but not completely positive maps, this
algorithm provides exponential advantages in entanglement detection and
quantification compared to protocols based on single-copy operations. In
addition, it facilitates the recovery of noiseless quantum states from multiple
copies of noisy states by implementing the inverse map of the corresponding
noise channel, offering an intriguing approach to handling quantum errors. Our
findings present a pathway for systematically and efficiently implementing
non-physical actions with quantum devices, thereby boosting the exploration of
potential quantum advantages across a wide range of information processing
tasks.
