Optimal encoding of oscillators into more oscillators

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.

30 pages, 13 figures. Final version accepted by Quantum, with typo corrected

References in corpus (11)

• Real-time quantum error correction beyond break-even
• A No-Go Theorem for Gaussian Quantum Error Correction
• Unitarily localizable entanglement of Gaussian states
• Mode-Wise Entanglement of Gaussian States
• Multi-mode bosonic Gaussian channels
• Distributed quantum sensing enhanced by continuous-variable error correction
• Encoding qubits in multimode grid states
• Not-so-normal mode decomposition
• Continuous-variable error correction for general Gaussian noises
• Qubit-oscillator concatenated codes: decoding formalism & code comparison
• Enhancing distributed sensing with imperfect error correction

Cited by in corpus (6)

• Entanglement-Based Quantum Information Technology
• Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications
• Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications
• Clifford operations and homological codes for rotors and oscillators
• Complexity of quantum tomography from genuine non-Gaussian entanglement
• Safeguarding Oscillators and Qudits with Distributed Two-Mode Squeezing