Improved Fault-Tolerance to Auxiliary Decay in the Quantum Error Correction of a Four-Mode Grid Code 

Bosonic codes encode a qubit in the large Hilbert space of a harmonic oscillator. The redundancy of the encoding can be used to correct errors, thereby leading to a more hardware-efficient approach to quantum computing. Notably, quantum error correction (QEC) above break-even in the single-mode grid code, known as the Gottesman-Kitaev-Preskill code [1], has been demonstrated experimentally [2]. 

Here, we numerically simulate for the first time quantum error correction in a four-mode grid code considering both cavity photon loss and auxiliary noise. We compare the code performance to the single-mode GKP code and the two-mode tesseract code [3]. Our numerical results show that the four-mode code has an improved performance below a certain cavity photon loss threshold and is also more robust to auxiliary noise. We explain the later by showing that the topological code property of path-connectedness in phase space can be linked to fault-tolerance to a single auxiliary decay event during the quantum error correction protocol [4]. We further outline how the concepts of simple connectedness or in general n-connectedness can lead to fault-tolerance to higher order auxiliary errors for codes with a higher number of modes. 

 

[1] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001) 

[2] V. V. Sivak et al., Nature 616, 55 (2023) 

[3] B. Royer, S. Singh, and S. M. Girvin, PRX Quantum 3, 010335 (2022) 

[4] B. Royer, S. Singh, and S. M. Girvin, Phys. Rev. Lett. 125, 260509 (2020)