Cost-Reduced All-Gaussian Universality with the Gottesman-Kitaev-Preskill Code: Resource-Theoretic Approach to Cost Analysis

TL;DR

This paper introduces a resource-efficient scheme for universal quantum computation with GKP codes, reducing non-Gaussian resource requirements by directly preparing the magic state and analyzing transformation bounds.

Contribution

It proposes a novel method to implement universal quantum computation with GKP codes without magic state distillation, utilizing a resource-theoretic approach for cost analysis.

Findings

Achieves reduction in non-Gaussian resource requirements.

Provides bounds on transformations between GKP states.

Demonstrates cost-effective fault-tolerant quantum computation.

Abstract

The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged as a key technique in achieving fault-tolerant quantum computation using photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502 (2019)] showed that experimentally tractable Gaussian operations combined with preparing a GKP codeword $\lvert 0\rangle$ suffice to implement universal quantum computation, this implementation scheme involves a distillation of a logical magic state $\lvert H\rangle$ of the GKP code, which inevitably imposes a trade-off between implementation cost and fidelity. In contrast, we propose a scheme of preparing $\lvert H\rangle$ directly and combining Gaussian operations only with $\lvert H\rangle$ to achieve the universality without this magic state distillation. In addition, we develop an analytical method to obtain bounds of fundamental limit on transformation between $\lvert H\rangle$ and $\lvert 0\rangle$, finding an application of quantum resource theories to cost analysis of quantum computation with the GKP code. Our results lead to an essential reduction of required non-Gaussian resources for photonic fault-tolerant quantum computation compared to the previous scheme.