Optimal encoding of oscillators into more oscillators

Bosonic encoding of quantum information into harmonic oscillators is a hardware effi-cient approach to battle noise. In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic enco d-ing, but also extend the applicability of er -ror correction to continuous-variable states ubiquitous in quantum sensing and commu-nication. In this work, we derive the op-timal 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 er -ror can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this opti-mal code design problem can be efficiently solved, and we further provide numerical evi-dence 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 di-mensional lattices. As a by-pro duct, 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.

Automated summary

In this work, we derive the optimal oscillator-to-oscillator codes for homogeneous noise, including the D4 lattice encoding for two-mode data and ancilla, which outperforms the product of lower dimensional lattices. Additionally, we prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding.