GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuous variable logical qubits that can be corrected for errors caused by phase space displacements. Their experimental realization is challenging, in particular, using propagating fields, where quantum information is encoded in the quadratures of the electromagnetic field. However, traveling photons are essential in many applications of GKP codes involving the long-distance transmission of quantum information. We introduce a new method for encoding GKP states in propagating fields using single photons, each occupying a distinct auxiliary mode given by the propagation direction. The GKP states are defined as highly correlated states described by collective continuous modes, as time and frequency. We analyze how the error detection and correction protocol scales with the total photon number and the spectral width. We show that the obtained code can be corrected for displacements in time-frequency phase space, which correspond to dephasing, or rotations, in the quadrature phase space and to photon losses. Most importantly, we show that generating two-photon GKP states is relatively simple, and that such states are currently produced and manipulated in several photonic platforms where frequency and time-bin biphoton entangled states can be engineered.

中文翻译：

---

单光子连续模态变量的 Gottesman-Kitaev-Preskill 编码

GKP 态由 Gottesman、Kitaev 和 Preskill 引入，是连续变量逻辑量子位，可以纠正由相空间位移引起的错误。他们的实验实现具有挑战性，特别是使用传播场，其中量子信息被编码在电磁场的正交中。然而，行进光子在许多涉及量子信息长距离传输的 GKP 码应用中至关重要。我们引入了一种使用单光子在传播场中编码 GKP 态的新方法，每个光子占据由传播方向给定的不同辅助模式。 GKP 状态被定义为由集体连续模式描述的高度相关状态，如时间和频率。我们分析了错误检测和校正协议如何随总光子数和光谱宽度进行缩放。我们表明，所获得的代码可以针对时频相空间中的位移进行校正，这对应于正交相空间中的移相或旋转以及光子损失。最重要的是，我们表明生成双光子 GKP 态相对简单，并且这种态目前在几个光子平台中产生和操纵，在这些平台中可以设计频率和时间仓双光子纠缠态。