Chasing shadows with Gottesman-Kitaev-Preskill codes

• URL: http://arxiv.org/abs/2411.00235v1
• Date: Thu, 31 Oct 2024 22:16:06 GMT
• Title: Chasing shadows with Gottesman-Kitaev-Preskill codes
• Authors: Jonathan Conrad, Jens Eisert, Steven T. Flammia,
• Abstract summary: In this work, we consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. In particular, we construct a logical shadow tomography protocol via twirling of CV-POVMs by displacement operators and Gaussian unitaries. For photon-parity measurements, logical GKP shadow tomography is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling schemes.
• Score: 0.3277163122167433
• License:
• Abstract: The infinitude of the continuous variable (CV) phase space is a serious obstacle in designing randomized tomography schemes with provable performance guarantees. A typical strategy to circumvent this issue is to impose a regularization, such as a photon-number cutoff, to enable the definition of ensembles of random unitaries on effective subspaces. In this work, we consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. In particular, we construct a logical shadow tomography protocol via twirling of CV-POVMs by displacement operators and Gaussian unitaries. In the special case of heterodyne measurement, the shadow tomography protocol yields a probabilistic decomposition of any input state into Gaussian states that simulate the encoded logical information of the input relative to a fixed GKP code and we prove bounds on the Gaussian compressibility of states in this setting. For photon-parity measurements, logical GKP shadow tomography is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling schemes and finally, using the existence of a Haar measure over symplectic lattices, we derive a Wigner sampling scheme via random GKP codes. This protocol establishes, via explicit sample complexity bounds, how Wigner samples of any input state from random points relative to a random GKP codes can be used to estimate any sufficiently bounded observable on CV space.

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