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The dependence of quantum algorithms on state fidelity is difficult to characterize analytically and is best explored experimentally as hardware scales and noisy simulations become intractable. While low fidelity states are often disregarded, they may still retain valuable information, as long as their dominant eigenvector approximates the target state. Through classical purification, we demonstrate the ability to recover resources specific to quantum computing such as magic and entanglement from noisy states generated by Pauli product formulas, which are common subroutines of many quantum algorithms. The fidelity of purified states represents the noise floor of a given computation and we show its dependence on both the magnitude and order in which magic and entanglement are generated. Using an ion trap quantum device, we experimentally validate these findings by collecting classical shadow data for a range of small circuits. While overall consistent, our results reveal meaningful differences in the captured qubit correlations, further highlighting the gap between conventional numerical studies and real experimental outcomes. In both simulation and experiment, we show the advantage of designing methods targeting states which are more robust against noise. This study uses quantum informatic tools for analyzing quantum algorithms in a noisy framework, and demonstrates practical strategies for optimizing quantum circuit performance.
