Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here we prove several results on efficiently learning Pauli channels, and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on n qubits to a fixed relative precision with O(n 2^n) measurements. For a Pauli channel with only s nonzero error rates, we show how to learn it with only O(s n) measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most k-local correlations, we can learn an entire n-qubit Pauli channel with only O(n^2 log n) measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device. Joint work with Robin Harper, Joel Wallman, and Wenjun Yu, arXiv:1907.13022, arXiv:1907.12976. 

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