Bridging magic and non-Gaussian resources via Gottesman-Kitaev-Preskill encoding

TL;DR

This paper establishes a fundamental connection between non-Gaussian states in continuous-variable systems and magic states in discrete-variable systems through GKP encoding, linking their resource measures and implications for quantum computation.

Contribution

It introduces a magic measure based on Wigner function negativity for GKP-encoded states and demonstrates the necessity of non-Gaussian operations for certain logical gates.

Findings

Negativity of the continuous-variable Wigner function matches a discrete magic measure.

Provides a continuous-variable representation of stabilizer Rénnyi entropy.

Shows non-Gaussian operations are required for implementing multi-qubit non-Clifford gates in GKP codes.

Abstract

Although the similarity between non-stabilizer states -- also known as magic states -- in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer R\'enyi entropy -- a recent proposal for a magic measure for multi-qubit states. With this in hand, we give a classical simulation algorithm with runtime scaling with the resource contents, quantified by our magic measures. We also employ our results to prove that implementing a multi-qubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.