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

TL;DR

This paper introduces a new algorithm to measure Stabilizer Rnyi Entropies of unknown quantum states by encoding entropy into purity, enabling efficient resource quantification for quantum computation.

Contribution

The work presents an alternative method for estimating Stabilizer Rnyi Entropies using purity measurements, including a channel construction and benchmarking for qubits.

Findings

The algorithm works for all integers >1.

It demonstrates a relationship between non-stabilizerness and entanglement.

A resource hiding phenomenon is identified.

Abstract

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.