Quantum error correction with the color-Gottesman-Kitaev-Preskill code

Abstract

The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum errorcorrecting code. Since the GKP code only protects against small shift errors in p and q quadratures, it is necessary to concatenate the GKP code with a stabilizer code for the larger error correction. In this paper, we consider the concatenation of the single-mode GKP code with the two-dimension (2D) color code (color-GKP code) on the square-octagon lattice. We use the Steane type scheme with a maximum-likelihood estimation (ME-Steane scheme) for GKP error correction and show its advantage for the concatenation. In our main work, the minimum-weight perfect matching (MWPM) algorithm is applied to decode the color-GKP code. Complemented with the continuous-variable information from the GKP code, the threshold of 2D color code is improved. If only data GKP qubits are noisy, the threshold reaches σ ≈ 0.59 (p ≈ 13.3%) compared with p = 10.2% of the normal 2D color code. If measurements are also noisy, we introduce the generalized Restriction Decoder on the three-dimension space-time graph for decoding. The threshold reaches σ ≈ 0.46 when measurements in the GKP error correction are noiseless, and σ ≈ 0.24 when all measurements are noisy. Lastly, the good performance of the generalized Restriction Decoder is also shown on the normal 2D color code giving the threshold at 3.1% under the phenomenological error model.