Chasing shadows with Gottesman-Kitaev-Preskill codes
Abstract: 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. We construct a logical shadow tomography protocol via twirling of continuous variable 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 we derive a Wigner sampling scheme via random GKP codes. This protocol establishes how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable on CV space. This construction shows how a description of the physical state of the system can be reconstructed from encoded logical information relative to a random code and further highlights the power of performing idealized GKP error correction as a tomographic resource.
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