Efficient sampling for Pauli-measurement-based shadow tomography in direct fidelity estimation

A constant number of random Clifford measurements allows the classical shadow protocol to perform direct fidelity estimation (DFE) with high precision. However, estimating properties of an unknown quantum state is expected to be more feasible with random Pauli measurements than with random Clifford measurements in the near future. Inspired by the importance sampling technique applied to sampling Pauli measurements for DFE, we show that similar strategies can be derived from classical shadows. Specifically, we describe efficient methods using only local Pauli measurements to perform DFE with Greenberger-Horne-Zeilinger (GHZ), <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>W</a:mi></a:mrow></a:math>, and Dicke states, establishing tighter bounds (by factors of 14.22 and 16 for GHZ and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>W</b:mi></b:mrow></b:math>/Dicke, respectively) on the number of measurements required for desired precision. These protocols are derived by adjusting the distribution of observables. Notably, they require no preprocessing steps other than the sampling algorithms.