  The manipulation of quantum states through linear maps beyond quantum
operations has many important applications in various areas of quantum
information processing. Current methods simulate unphysical maps by sampling
physical operations, but in a classical way. In this work, we show that using
quantum measurement in place of classical sampling leads to lower simulation
costs for general Hermitian-preserving maps. Remarkably, we establish the
equality between the simulation cost and the well-known diamond norm, thus
closing a previously known gap and assigning diamond norm a universal
operational meaning as a map's simulability. We demonstrate our method in two
applications closely related to error mitigation and quantum machine learning,
where it exhibits a favorable scaling. These findings highlight the power of
quantum measurement in simulating unphysical operations, in which quantum
interference is believed to play a vital role. Our work paves the way for more
efficient sampling techniques and has the potential to be extended to more
quantum information processing scenarios.
