We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous, or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators in terms of five integer indices. On the basis of these indices, it can be decided whether the half-step operator can be obtained from a continuous Hamiltonian driving, or not. The half-step operator determines two Floquet operators, obtained by starting the driving at zero or at half-period, respectively. These are called timeframes and are chiral symmetric quantum walks. Conversely, we show under which conditions two chiral symmetric walks determine a common half-step operator. Moreover, we clarify the connection between the classification of half-step operators and the corresponding quantum walks. Within this theory, we prove bulk-edge correspondence and show that a second timeframe allows to distinguish between symmetry protected edge states at \(+\,1\) and \(-\,1\) which is not possible for a single timeframe.

我们将一维晶格上的手性对称周期性驱动量子系统分类。驱动过程是局部的，可以是连续的，也可以是时间上的离散，并且我们假设相应的Floquet运算符存在间隙条件。分析是根据单一运算符在半周期，半步运算符进行的。我们根据五个整数索引对半步算子的连接类进行了完整分类。基于这些指标，可以确定是否可以从连续的汉密尔顿式驾驶中获得半步操。半步运算符确定两个Floquet运算符，分别通过在零周期或半周期开始驱动获得。这些被称为时间范围，是手性对称的量子步态。反过来，我们展示了在哪些条件下两个手性对称步态决定了一个共同的半步算子。此外，我们阐明了半步算子的分类与相应的量子游动之间的联系。在该理论中，我们证明了体边缘对应关系，并表明第二个时间范围可以区分在以下位置受对称保护的边缘状态\（+ \，1 \）和\（-\，1 \），这在单个时间范围内是不可能的。