A promising route towards fault-tolerant quantum error correction is the concatenation of a Gottesman-Kitaev-Preskill (GKP) code with a qubit code. Development of such concatenated codes requires simulation tools which realistically model noise, while being able to simulate the dynamics of many modes. However, so far, large-scale simulation tools for concatenated GKP codes have been limited to idealized noise models and GKP code implementations. Here, we introduce the Bosonic Pauli+ model (BP+), which can be simulated efficiently for a large number of modes, while capturing the rich dynamics in the bosonic multi-mode Hilbert space. We demonstrate the method by simulating a hybrid surface code, where the data qubits are finite-energy GKP qubits stabilized using the small-Big-small (sBs) protocol, and the syndrome qubits are standard two-level systems. Using BP+, we present logical error rates of such an implementation. Confidence in the accuracy of the method is gained by comparing its predictions with full time evolution simulations for several relevant quantum circuits. While developed specifically for GKP qubits stabilized using the sBs protocol, the mathematical structure of BP+ is generic and may be applicable also to the simulation of concatenations using other bosonic codes.

中文翻译：

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Bosonic Pauli+：级联 Gottesman-Kitaev-Preskill 码的高效仿真


实现容错量子纠错的一条有前途的途径是将 Gottesman-Kitaev-Preskill （GKP） 代码与量子比特代码连接起来。开发这种级联代码需要仿真工具，这些工具能够真实地对噪声进行建模，同时能够仿真多种模式的动力学。然而，到目前为止，用于级联 GKP 代码的大规模仿真工具仅限于理想化噪声模型和 GKP 代码实现。在这里，我们介绍了 Bosonic Pauli+ 模型 （BP+），该模型可以针对大量模态进行高效仿真，同时捕捉玻色子多模态 Hilbert 空间中的丰富动力学。我们通过模拟混合表面代码来演示该方法，其中数据量子比特是使用小-大-小 （sBs） 协议稳定的有限能量 GKP 量子比特，而综合征量子比特是标准的两级系统。使用 BP+，我们展示了这种实现的逻辑错误率。通过将其预测与几个相关量子电路的全时演化模拟进行比较，可以获得对该方法准确性的信心。虽然 BP+ 的数学结构是专门为使用 sBs 协议稳定的 GKP 量子比特开发的，但它是通用的，也可能适用于使用其他玻色子码的串联模拟。