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In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic encoding, but also extend the applicability of error correction to continuous-variable states ubiquitous in quantum sensing and communication. In this work, we derive the optimal oscillator-to-oscillator codes among the general family of Gottesman-Kitaev-Preskill (GKP)-stablizer codes for homogeneous noise. We prove that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing (TMS) code. The optimal encoding to minimize the geometric mean error can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this optimal code design problem can be efficiently solved, and we further provide numerical evidence that a hexagonal GKP lattice is optimal and strictly better than the previously adopted square lattice. For the multimode case, general GKP lattice optimization is challenging. In the two-mode data and ancilla case, we identify the D4 lattice\u2014a 4-dimensional dense-packing lattice\u2014to be superior to a product of lower dimensional lattices. As a by-product, the code reduction allows us to prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding, even when the ancilla are not GKP states.<\/jats:p>","DOI":"10.22331\/q-2023-08-16-1082","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T10:07:50Z","timestamp":1692180470000},"page":"1082","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":7,"title":["Optimal encoding of oscillators into more oscillators"],"prefix":"10.22331","volume":"7","author":[{"given":"Jing","family":"Wu","sequence":"first","affiliation":[{"name":"James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA"}]},{"given":"Anthony J.","family":"Brady","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9554-3846","authenticated-orcid":false,"given":"Quntao","family":"Zhuang","sequence":"additional","affiliation":[{"name":"Ming Hsieh Department of Electrical and Computer Engineering & Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA"},{"name":"James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA"},{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]}],"member":"9598","published-online":{"date-parts":[[2023,8,16]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. R. Calderbank and Peter W. 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