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Chasing shadows with Gottesman-Kitaev-Preskill codes

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Chasing shadows with Gottesman-Kitaev-Preskill codes

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AbstractWe consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the 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 where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.Featured image: Illustration of Gaussian decomposition of arbitrary CV states via twirled heterodyne measurements. Relative to a GKP code described by a lattice L this shadow tomography protocol yields a probabilistic decomposition of the input state into Gaussian states that reproduce logical expectation values up to a logical depolarization M.Popular summaryThe continuous physical world we live in is vast; it is hard to control and understand. Focusing on discrete subsystems of a physical setup gives us so much control to prevent errors in our computation and allows us to tractably understand our systems as we develop their building blocks. For quantum computing, this task is facilitated through the use of bosonic codes, that distinguish a discrete quantum system within a continuous one. Famous bosonic codes are e.g. the trivial encoding, where qubit logic is encoded between a state of nothingness and the minimal something state of a continuous system. This is the structure typically found in, e.g., the popular transmon qubit that is being experimented on worldwide. Another interesting encoding is provided by the so-called Gottesman-Kiteav-Preskill (GKP) codes, where discrete information is hidden into the oﬀset of a periodic lattice structure that we endow the continuous variable (CV) space with. Now that we have broken it down, the state of a discrete system is easy to understand. There is only a finite number of measurements that need to be made so that every property of the system can be predicted. Easing the requirements even further, Huang, Kueng and Preskill have developed a protocol, the classical shadow, where a small set of random measurements already suﬃce to predict a large set of observables to high accuracy. In this protocol, however, it is important that the individual random measurements are distributed over the small qubit space well enough to cover it, at least in approximation. In this work we examine the equivalent problem for a continuous variable quantum system: we wish to use a small set of random measurements to understand the state to a suﬃcient degree that we can make sensible predictions. If we blindly tried to follow the previous strategy, it is clear where this breaks down: there is no obvious way to cover the infinite continuous variable space in an even manner with only a small (or even finite) number of measurements. We choose a particular route to challenge this problem. We pick a subspace of the CV space, which is prescribed by the lattice structure of a GKP code, and proceed with the eﬀective finite strategy. This gives us interesting insights on how the GKP code interacts with measurements on the CV system. What’s more is that now, we can go one-step further: using tricks from the theory of random lattices, we now randomize this procedure over the space of all possible lattices (all possible GKP codes) and show how the fixed-GKP code finite subspace tomography scheme can be lifted to a tomography protocol that scouts out all of the CV space. This procedure allows us to apply known statistical methods to now also benchmark how well arbitrary predictions can be made for CV states of arbitary structure. By using tools from the theory of random coding, this scheme shows how we can smoothly interpolate between understanding the logic of a continuous variable system, and its physics.► BibTeX data@article{Conrad2026chasingshadows, doi = {10.22331/q-2026-01-19-1973}, url = {https://doi.org/10.22331/q-2026-01-19-1973}, title = {Chasing shadows with {G}ottesman-{K}itaev-{P}reskill codes}, author = {Conrad, Jonathan and Eisert, Jens and Flammia, Steven T.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1973}, month = jan, year = {2026} }► References [1] D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, Bonilla A. J. P., N. Maskara, I. Cong, X. Gao, Pedro Sales R., T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin. ``Logical quantum processor based on reconfigurable atom arrays''. Nature 626, 58–65 (2023). https://doi.org/10.1038/s41586-023-06927-3 [2] V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret. ``Real-time quantum error correction beyond break-even''. Nature 616, 50–55 (2023). https://doi.org/10.1038/s41586-023-05782-6 [3] D. Gottesman, A. Kitaev, and J. Preskill. ``Encoding a qubit in an oscillator''. Phys. Rev. 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Dopico and Charles R. Johnson. ``Parametrization of the matrix symplectic group and applications''. SIAM J. Matrix Ana. Appl. 31, 650–673 (2009). https://doi.org/10.1137/060678221Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-19 16:48:37: Could not fetch cited-by data for 10.22331/q-2026-01-19-1973 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-19 16:48:38: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the 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 where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.Featured image: Illustration of Gaussian decomposition of arbitrary CV states via twirled heterodyne measurements. Relative to a GKP code described by a lattice L this shadow tomography protocol yields a probabilistic decomposition of the input state into Gaussian states that reproduce logical expectation values up to a logical depolarization M.Popular summaryThe continuous physical world we live in is vast; it is hard to control and understand. Focusing on discrete subsystems of a physical setup gives us so much control to prevent errors in our computation and allows us to tractably understand our systems as we develop their building blocks. For quantum computing, this task is facilitated through the use of bosonic codes, that distinguish a discrete quantum system within a continuous one. Famous bosonic codes are e.g. the trivial encoding, where qubit logic is encoded between a state of nothingness and the minimal something state of a continuous system. This is the structure typically found in, e.g., the popular transmon qubit that is being experimented on worldwide. Another interesting encoding is provided by the so-called Gottesman-Kiteav-Preskill (GKP) codes, where discrete information is hidden into the oﬀset of a periodic lattice structure that we endow the continuous variable (CV) space with. Now that we have broken it down, the state of a discrete system is easy to understand. There is only a finite number of measurements that need to be made so that every property of the system can be predicted. Easing the requirements even further, Huang, Kueng and Preskill have developed a protocol, the classical shadow, where a small set of random measurements already suﬃce to predict a large set of observables to high accuracy. In this protocol, however, it is important that the individual random measurements are distributed over the small qubit space well enough to cover it, at least in approximation. In this work we examine the equivalent problem for a continuous variable quantum system: we wish to use a small set of random measurements to understand the state to a suﬃcient degree that we can make sensible predictions. If we blindly tried to follow the previous strategy, it is clear where this breaks down: there is no obvious way to cover the infinite continuous variable space in an even manner with only a small (or even finite) number of measurements. We choose a particular route to challenge this problem. We pick a subspace of the CV space, which is prescribed by the lattice structure of a GKP code, and proceed with the eﬀective finite strategy. This gives us interesting insights on how the GKP code interacts with measurements on the CV system. What’s more is that now, we can go one-step further: using tricks from the theory of random lattices, we now randomize this procedure over the space of all possible lattices (all possible GKP codes) and show how the fixed-GKP code finite subspace tomography scheme can be lifted to a tomography protocol that scouts out all of the CV space. This procedure allows us to apply known statistical methods to now also benchmark how well arbitrary predictions can be made for CV states of arbitary structure. By using tools from the theory of random coding, this scheme shows how we can smoothly interpolate between understanding the logic of a continuous variable system, and its physics.► BibTeX data@article{Conrad2026chasingshadows, doi = {10.22331/q-2026-01-19-1973}, url = {https://doi.org/10.22331/q-2026-01-19-1973}, title = {Chasing shadows with {G}ottesman-{K}itaev-{P}reskill codes}, author = {Conrad, Jonathan and Eisert, Jens and Flammia, Steven T.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1973}, month = jan, year = {2026} }► References [1] D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, Bonilla A. J. P., N. Maskara, I. Cong, X. Gao, Pedro Sales R., T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. 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Chasing shadows with Gottesman-Kitaev-Preskill codes

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