Oscillator-to-oscillator codes do not have a threshold

TL;DR

Oscillator-to-oscillator codes, which use non-Gaussian states to protect quantum information, do not exhibit a threshold for error correction when using practical encoding and decoding methods, limiting their error suppression capabilities.

Contribution

This paper proves that physically implementable oscillator-to-oscillator codes cannot have a threshold, establishing fundamental limits on their error correction performance.

Findings

No threshold exists for these codes with practical encoding and decoding.

A lower bound on logical error probability depends only on squeezing, not code size.

Oscillator-to-oscillator codes cannot convert below-threshold physical noise into arbitrarily low logical noise.

Abstract

It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Noh et al. (PRL 125:080503, 2020) proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states. Here we ask if - in analogy to qubit error-correcting codes - there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold.