An Algorithm for Estimating $\alpha$-Stabilizer R\'enyi Entropies via Purity

Non-stabilizerness, or magic, is a resource for universal quantum computation in most fault-tolerant architectures; access to states with non-stabilizerness allows for non-classically simulable quantum computation to be performed. Quantifying this resource for unknown states is therefore essential to assessing their utility in quantum computation. The Stabilizer R\'enyi Entropies have emerged as a leading tool for achieving this, having already enabled one efficient algorithm for measuring non-stabilizerness. In addition, the Stabilizer R\'enyi Entropies have proven useful in developing connections between non-stabilizerness and other quantum phenomena. In this work, we introduce an alternative algorithm for measuring the Stabilizer R\'enyi Entropies of an unknown quantum state. Firstly, we show the existence of a state, produced from the action of a channel on $\alpha$ copies of some \ben{state $\rho$}, that encodes the $\alpha$-Stabilizer R\'enyi Entropy of $\rho$ into its purity. We detail several methods of applying this channel, and then, by employing existing purity-measuring algorithms, provide an algorithm for measuring the $\alpha$-Stabilizer R\'enyi Entropies for all integers $\alpha>1$. This algorithm is benchmarked for qubits and the resource requirements compared to other known algorithms. Finally, a non-stabilizerness/entanglement relationship is shown to exist in the algorithm, demonstrating a novel relationship between the two resources, before an instance of resource hiding is found.