Sample-Optimal Quantum Process Tomography with Non-Adaptive Incoherent Measurements

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 $\tilde{\mathcal{O}}(d_{\text{in}}^3{d}_{\text{out}}^3\mathrm{/}{\epsilon }^2)$ copies are sufficient to learn any quantum channel $C^{d_{\text{in}}\times d_{\text{in}}}\to C^{d_{\text{out}}\times d_{\text{out}}}$ to within $\epsilon$ in diamond norm. Moreover, we show that $\mathrm{\Omega }(d_{\text{in}}^3{d}_{\text{out}}^3\mathrm{/}{\epsilon }^2)$ copies are necessary for any strategy using incoherent non-adaptive measurements. This lower bound applies even for ancilla-assisted strategies.