Stabilization of finite-energy grid states of a quantum harmonic oscillator by reservoir engineering with two dissipation channels

TL;DR

This paper introduces an accessible Lindblad master equation to stabilize GKP grid states in a quantum harmonic oscillator, with applications in quantum error correction and metrology.

Contribution

It simplifies previous models for stabilizing GKP states, providing explicit energy estimates, convergence rates, and numerical simulations for state preparation.

Findings

Explicit energy bounds for solutions of the master equation

Estimated convergence rates to the codespace for GKP qubits

Numerical simulations demonstrating steady-state preparation of metrological states

Abstract

We propose and analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator, simplifying a previous proposal to alleviate implementation constraints. It approximately stabilizes periodic grid states introduced in 2001 by Gottesman, Kitaev and Preskill (GKP), with applications for quantum error correction and quantum metrology. We obtain explicit estimates for the energy of the solutions of the Lindblad master equation. We estimate the convergence rate to the codespace when stabilizing a GKP qubit, and numerically study the effect of noise. We then present simulations illustrating how a modification of parameters allows preparing states of metrological interest in steady-state.