Observation of a Fault Tolerance Threshold with Concatenated Codes

TL;DR

This paper demonstrates a fault-tolerance threshold in quantum error correction using concatenated codes and tensor network decoders, validated on ion-trap hardware and applicable to quantum memory and computation.

Contribution

It introduces a new fault-tolerant protocol with high noise thresholds and low overhead, utilizing tensor network decoders for effective error correction.

Findings

Achieved a state preparation threshold of approximately 8.9% for erasure errors.

Demonstrated the protocol's effectiveness on ion-trap hardware with added noise.

Validated the scheme's potential for quantum memory and universal quantum computation.

Abstract

We introduce a fault-tolerant protocol for code concatenation of a generalized Shor code using a butterfly network architecture with high noise thresholds and low ancilla overhead to allow implementation on current devices. We develop a probability passing decoder using tensor networks that applies Bayesian updates to the marginal error probabilities after each layer of checks, achieving a state preparation threshold of $e_c \approx 0.089$ for erasure errors, and $\approx 0.015$ for unheralded noise. We implement our state preparation protocol on ion-trap hardware with added noise to demonstrate the threshold behavior in a real quantum device. We further theoretically test the performance of our scheme as a quantum memory and for universal quantum computation through the preparation of low-noise magic states for state distillation and $T$-gate injection.