Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes

TL;DR

This paper advances the understanding of approximate GKP states for quantum computing by analyzing their imperfections, proposing a state preparation method, and evaluating their practicality for NISQ devices.

Contribution

It provides a detailed analysis of imperfect GKP states' impact on quantum circuits and introduces a photonic state preparation approach for GKP states.

Findings

Imperfect GKP states' errors can be effectively tracked and mitigated in quantum circuits.

Photonic state preparation can produce GKP states with nearly equal resource requirements.

Numerical results show feasibility of generating GKP states in the Bloch sphere with practical resources.

Abstract

Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behaviour of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently-developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (Noisy Intermediate-Scale Quantum) devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.