  The quantum channel decomposition techniques, which contain the so-called
probabilistic error cancellation and gate/wire cutting, are powerful approach
for simulating a hard-to-implement (or an ideal) unitary operation by
concurrently executing relatively easy-to-implement (or noisy) quantum
channels. However, such virtual simulation necessitates an exponentially large
number of decompositions, thereby significantly limiting their practical
applicability. This paper proposes a channel decomposition method for target
unitaries that have their input and output conditioned on specific quantum
states, namely unitaries with pre- and post-selection. Specifically, we
explicitly determine the requisite number of decomposing channels, which could
be significantly smaller than the selection-free scenario. Furthermore, we
elucidate the structure of the resulting decomposed unitary. We demonstrate an
application of this approach to the quantum linear solver algorithm,
highlighting the efficacy of the proposed method.
