# Fault-tolerant preparation of approximate GKP states

## TL;DR

This paper develops a fault-tolerant protocol for preparing approximate GKP states, crucial for quantum error correction, and analyzes its implementation challenges and robustness under realistic noise conditions.

## Contribution

It introduces a rigorous fault-tolerance definition for GKP state preparation and proposes a fault-tolerant phase estimation protocol suitable for circuit QED.

## Key findings

- The protocol uses one flag qubit and selective acceptance to prevent large shift errors.
- Analytic expressions describe effects of non-linear dispersive shift and Kerr non-linearity.
- Numerical results show accepted states retain error correction capabilities despite noise.

## Abstract

Gottesman-Kitaev-Preskill (GKP) states appear to be amongst the leading candidates for correcting errors when encoding qubits into oscillators. However the preparation of GKP states remains a significant theoretical and experimental challenge. Until now, no clear definitions for fault-tolerantly preparing GKP states have been provided. Without careful consideration, a small number of faults can lead to large uncorrectable shift errors. After proposing a metric to compare approximate GKP states, we provide rigorous definitions of fault-tolerance and introduce a fault-tolerant phase estimation protocol for preparing such states. The fault-tolerant protocol uses one flag qubit and accepts only a subset of states in order to prevent measurement readout errors from causing large shift errors. We then show how the protocol can be implemented using circuit QED. In doing so, we derive analytic expressions which describe the leading order effects of the non-linear dispersive shift and Kerr non-linearity. Using these expressions, we show that to mitigate the non-linear dispersive shift and Kerr terms would require the protocol to be implemented on time scales four orders of magnitude longer than the time scales relevant to the protocol for physically motivated parameters. Despite these restrictions, we numerically show that a subset of the accepted states of the fault-tolerant phase estimation protocol maintain good error correcting capabilities even in the presence of noise.