# Efficient estimation of Pauli channels

## TL;DR

This paper presents efficient methods for learning and estimating Pauli channels in quantum systems, enabling accurate characterization of noise and errors with fewer measurements, which is crucial for quantum computing reliability.

## Contribution

The paper introduces new measurement-efficient algorithms for estimating Pauli channels and their error rates, including channels with local correlations, advancing quantum noise characterization.

## Key findings

- Achieves high-probability estimation of n-qubit Pauli channels with O(ε^{-2} n 2^n) measurements.
- Provides measurement bounds for estimating s Pauli error rates with O(ε^{-4} log s log s/ε).
- Learns channels with k-local correlations using O_k(ε^{-2} n^2 log n) measurements, efficient in qubit number.

## Abstract

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 with high probability to a relative precision $\epsilon$ using $O\bigl(\epsilon^{-2} n 2^n\bigr)$ measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next we show that the error rates for an arbitrary set of $s$ Pauli errors can be estimated to a relative precision $\epsilon$ using $O\bigl(\epsilon^{-4} \log s\log s/\epsilon\bigr)$ 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 to relative precision $\epsilon$ with only $O_k\bigl(\epsilon^{-2} n^2 \log n \bigr)$ 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.