Optimal encoding of oscillators into more oscillators

• URL: http://arxiv.org/abs/2212.11970v4
• Date: Wed, 22 Nov 2023 21:11:09 GMT
• Title: Optimal encoding of oscillators into more oscillators
• Authors: Jing Wu, Anthony J. Brady and Quntao Zhuang
• Abstract summary: We show that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing code. For single-mode data and ancilla, this optimal code design problem can be efficiently solved. We identify the D4 lattice -- a 4-dimensional dense-packing lattice -- to be superior to a product of lower dimensional lattices.
• Score: 5.717368673366845
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Bosonic encoding of quantum information into harmonic oscillators is a hardware efficient approach to battle noise. 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 -- a 4-dimensional dense-packing lattice -- to 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.

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