# All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code

## TL;DR

This paper demonstrates that Gaussian operations combined with GKP-encoded Pauli eigenstates enable universal, fault-tolerant quantum computing without the need for non-Gaussian elements, relying on sufficient squeezing and low noise.

## Contribution

It shows that applying GKP error correction to Gaussian states produces magic states, establishing universality with only Gaussian operations under certain conditions.

## Key findings

- Gaussian input states yield distillable magic states after GKP error correction
- Fault tolerance achievable with sufficient squeezing and low external noise
- Gaussian operations suffice for universal quantum computing with GKP codes

## Abstract

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.