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Quantum error correction (QEC) plays an essential role in fault-tolerantlyrealizing quantum algorithms of practical interest. Among different approachesto QEC, encoding logical quantum information in harmonic oscillator modes hasbeen shown to be promising and hardware efficient. In this work, we studymultimode Gottesman-Kitaev-Preskill (GKP) codes, encoding a qubit in manyoscillators, through a lattice perspective. In particular, we implement aclosest point decoding strategy for correcting random Gaussian shift errors.For decoding a generic multimode GKP code, we first identify its correspondinglattice followed by finding the closest lattice point in its symplectic duallattice to a candidate shift error compatible with the error syndrome. We usethis method to characterize the error correction capabilities of several knownmultimode GKP codes, including their code distances and fidelities. We alsoperform numerical optimization of multimode GKP codes up to ten modes and findthree instances (with three, seven and nine modes) with better code distancesand fidelities compared to the known GKP codes with the same number of modes.While exact closest point decoding incurs exponential time cost in the numberof modes for general unstructured GKP codes, we give several examples ofstructured GKP codes (i.e., of the repetition-rectangular GKP code types) wherethe closest point decoding can be performed exactly in linear time. For thesurface-GKP code, we show that the closest point decoding can be performedexactly in polynomial time with the help of a minimum-weight-perfect-matchingalgorithm (MWPM). We show that this MWPM closest point decoder improves boththe fidelity and the noise threshold of the surface-GKP code to 0.602 comparedto the previously studied MWPM decoder assisted by log-likelihood analoginformation which yields a noise threshold of 0.599.

Closest lattice point decoding for multimode Gottesman-Kitaev-Preskill codes