Abstract
How many copies of a quantum process are necessary and sufficient to construct an approximate classical description of it? We extend the result of Surawy-Stepney, Kahn, Kueng, and Guta (2022) to show that O(d_in³d_out³/ε²) copies are sufficient to learn any quantum channel C^d_in× d_in → C^d_out× d_out to within ε in diamond norm. Moreover, we show that Ω(d_in³ d_out³/ε²) copies are necessary for any strategy using incoherent non-adaptive measurements. This lower bound applies even for ancilla-assisted strategies.