Quantum Resources for Pure Thermal Shadows

Calculating the properties of Gibbs states is an important task in Quantum Chemistry and Quantum Machine Learning. Previous work has proposed a quantum algorithm which predicts Gibbs state expectation values for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> observables from only log <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M$</tex> measurements, by combining classical shadows and quantum signal processing for a new estimator called Pure Thermal Shadows. In this work, we perform resource analysis for the circuits used in this algorithm, finding that quantum signal processing contributes most significantly to gate count and depth as system size increases. The implementation we use for this also features an improvement to the algorithm in the form of more efficient random unitary generation steps. Moreover, given the ramifications of the resource analysis, we argue that its potential utility could be constrained to Fault Tolerant devices sampling from the Gibbs state of a large, cool system.