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

• URL: http://arxiv.org/abs/2406.06418v4
• Date: Wed, 05 Mar 2025 01:59:49 GMT
• Title: Bridging magic and non-Gaussian resources via Gottesman-Kitaev-Preskill encoding
• Authors: Oliver Hahn, Giulia Ferrini, Ryuji Takagi,
• Abstract summary: We establish a fundamental link between non-stabilizer states and non-Gaussian states in continuous-variable systems.<n>We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure.<n>We also provide a continuous-variable representation of the stabilizer R'enyi entropy.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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