We construct stabilizer states and error-correcting codes on combinations of discrete- and continuous-variable systems, generalizing the Gottesman-Kitaev-Preskill (GKP) quantum lattice formalism. Our framework absorbs the discrete phase space of a qudit into a hybrid phase space parameterizable entirely by the continuous variables of a harmonic oscillator. The unit cell of a hybrid quantum lattice grows with the qudit dimension, yielding a way to simultaneously measure an arbitrarily large range of non-commuting position and momentum displacements. Simple hybrid states can be obtained by applying a conditional displacement to a Gottesman-Kitaev-Preskill (GKP) state and a Pauli eigenstate, or by encoding some of the physical qudits of a stabilizer state into a GKP code. The states' oscillator-qudit entanglement cannot be generated using symplectic (i.e., Gaussian-Clifford) operations, distinguishing them as a resource from tensor products of oscillator and qudit stabilizer states. Simple hybrid codes can be thought of as subsystem GKP codes whose gauge factor is entangled with a qudit. Our numerical investigations suggest that such codes can sometimes outperform GKP codes against physical noise, and their decoders can be tuned to accommodate either more qudit or more oscillator errors. We also relate stabilizer codes to non-commutative tori, identifying that a general construction of such tori yields multi-mode multi-qudit extensions of GKP codes. We explicitly calculate these codes' logical dimension and logical operators by utilizing the Morita equivalence between their stabilizer and logical tori. We provide examples using commutation matrices, integer symplectic matrices, and binary codes.

翻译：我们构建了离散与连续变量系统组合上的稳定子态与纠错码，推广了Gottesman-Kitaev-Preskill（GKP）量子晶格形式体系。我们的框架将量子比特的离散相空间吸收进一个可完全由谐振子连续变量参数化的混合相空间。混合量子晶格的单位晶胞随量子比特维度增长，从而提供了一种同时测量任意大范围非对易位置与动量位移的方法。简单混合态可通过以下方式获得：对GKP态与泡利本征态施加条件位移，或将稳定子态的部分物理量子比特编码为GKP码。这些态的振子-量子比特纠缠无法通过辛操作（即高斯-克利福德操作）生成，这使其区别于振子与量子比特稳定子态的张量积而成为一种资源。简单混合码可视为子系统GKP码，其规范因子与量子比特纠缠。我们的数值研究表明，此类码在应对物理噪声时有时能超越GKP码，且其解码器可调整以适应更多量子比特误差或更多振子误差。我们还将稳定子码与非交换环面相关联，指出此类环面的一般构造可产生GKP码的多模多量子比特扩展。通过利用其稳定子环面与逻辑环面间的森田等价，我们显式计算了这些码的逻辑维度与逻辑算符。我们使用对易矩阵、整数辛矩阵与二进制码提供了具体示例。