Limitations of local update recovery in stabilizer-GKP codes: a quantum optimal transport approach

TL;DR

This paper establishes an upper bound on the fault-tolerance threshold for local update recovery in GKP-stabilizer codes under photon loss noise, using a novel bosonic Wasserstein distance to analyze continuous variable quantum systems.

Contribution

It introduces a continuous-variable quantum Wasserstein distance and applies it to derive bounds on fault-tolerance thresholds in GKP-stabilizer codes with local recovery.

Findings

Encoded information is lost exponentially above the threshold loss rate.

The bosonic Wasserstein distance captures locality in CV quantum systems.

The approach extends discrete quantum results to continuous variable systems.

Abstract

Local update recovery seeks to maintain quantum information by applying local correction maps alternating with and compensating for the action of noise. Motivated by recent constructions based on quantum LDPC codes in the finite-dimensional setting, we establish an analytic upper bound on the fault-tolerance threshold for concatenated GKP-stabilizer codes with local update recovery. Our bound applies to noise channels that are tensor products of one-mode beamsplitters with arbitrary environment states, capturing, in particular, photon loss occurring independently in each mode. It shows that for loss rates above a threshold given explicitly as a function of the locality of the recovery maps, encoded information is lost at an exponential rate. This extends an early result by Razborov from discrete to continuous variable (CV) quantum systems. To prove our result, we study a metric on bosonic states akin to the Wasserstein distance between two CV density functions, which we call the bosonic Wasserstein distance. It can be thought of as a CV extension of a quantum Wasserstein distance of order 1 recently introduced by De Palma et al. in the context of qudit systems, in the sense that it captures the notion of locality in a CV setting. We establish several basic properties, including a relation to the trace distance and diameter bounds for states with finite average photon number. We then study its contraction properties under quantum channels, including tensorization, locality and strict contraction under beamsplitter-type noise channels. Due to the simplicity of its formulation, and the established wide applicability of its finite-dimensional counterpart, we believe that the bosonic Wasserstein distance will become a versatile tool in the study of CV quantum systems.