  How well can we approximate a quantum channel output state using a random
codebook with a certain size? In this work, we study the quantum soft covering
problem. Namely, we use a random codebook with codewords independently sampled
from a prior distribution and send it through a classical-quantum channel to
approximate the target state. When using a random codebook sampled from an
independent and identically distributed prior with a rate above the quantum
mutual information, we show that the expected trace distance between the
codebook-induced state and the target state decays with exponent given by the
sandwiched R\'enyi information. On the other hand, when the rate of the
codebook size is below the quantum mutual information, the trace distance
converges to one exponentially fast. We obtain similar results when using a
random constant composition codebook, whereas the sandwiched Augustin
information expresses the error exponent. In addition to the above large
deviation analysis, our results also hold in the moderate deviation regime.
That is, we show that even when the rate of the codebook size approaches the
quantum mutual information moderately quickly, the trace distance still
vanishes asymptotically.
