Minimum energy decoding for multimode Gottesman-Kitaev-Preskill (GKP) codes

Quantum error correction (QEC) plays an essential role in fault-tolerantly realizing quantum algorithms of practical interest. Among different approaches to QEC, encoding logical quantum information in harmonic oscillator modes has been shown to be promising and hardware efficient. In this work, we consider multimode Gottesman-Kitaev-Preskill (GKP) codes (encoding a qubit in many oscillators) from a lattice perspective and study a minimum-energy decoding strategy for correcting random Gaussian shift errors. For a given GKP code, we first identify its corresponding lattice, followed by considering the Voronoi cell structure of its symplectic dual lattice. The minimum-energy decoder works by finding the closest point in the dual lattice to a given shift error. We also use the same method to compute the code distances of a multimode GKP code. While minimum energy decoding incurs exponential time cost in the number of modes for general unstructured GKP codes, we give several examples of structured GKP codes where the minimum energy decoding can be performed in polynomial time.