Here, I prove Lemma 3.3 of Watrous (2018) in some detail and point out the neat technique used to cast a sum of operators as a product of operators in a larger, augmenting space, which mixes quantum states and amplitudes. All references in this section refer to Watrous (2018).

Given the sampling error, how close to true is our estimate of the derived statistic from the bitstrings? How does the sampling error in the bitstrings propagate through the computation? How many M shots should an experimentalist run to obtain a certain confidence?