Adaptivity is not helpful for Pauli channel learning

We prove that adaptive strategies offer no advantage over non-adaptive ones for learning and testing Pauli channels using entangled inputs. This key observation allows us to characterize the query complexity for several fundamental tasks by translating optimal classical estimation algorithms into the quantum setting. First, we determine the tight query complexity for learning a Pauli channel under the general <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&amp;#x2113;</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math> norm, providing results that improve upon or match the best-known bounds for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&amp;#x2113;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>&amp;#x2113;</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&amp;#x2113;</mml:mi><mml:mi mathvariant="normal">&amp;#x221E;</mml:mi></mml:msub></mml:math> distances. Second, we resolve the complexity of testing whether a Pauli channel is a white noise source. Finally, we show that the optimal query complexities for estimating the Shannon entropy and support size of the channel's error distribution, and for estimating the diamond distance between two Pauli channels, are all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">&amp;#x0398;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mstyle displaystyle="false" scriptlevel="0"><mml:mfrac><mml:msup><mml:mn>4</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>)</mml:mo></mml:mrow></mml:math>.