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

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 Rényi 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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Cited by in corpus (8)

• Stabilizer Rényi Entropy and Conformal Field Theory
• Classical simulation and quantum resource theory of non-Gaussian optics
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space
• Steganographic Entanglement Sharing
• Classical simulation of circuits with realistic odd-dimensional Gottesman-Kitaev-Preskill states
• Symmetric channel verification for purifying noisy quantum channels
• Tessellation codes: encoded quantum gates by geometric rotation
• Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing