Analytical solutions to the time-dependent Schrodinger equation describing a
driven two-level system are invaluable to many areas of physics, but they are
also extremely rare. Here, we present a simple algorithm that generates an
unlimited number of exact analytical solutions. We show that a general
single-axis driving term and its corresponding evolution operator are
determined by a single real function which is constrained only by a certain
inequality and initial conditions. Any function satisfying these constraints
yields an exact analytical solution. We demonstrate this method by presenting
several new exact solutions to the time-dependent Schrodinger equation. Our
general method and many of the new solutions we present are particularly
relevant to qubit control in quantum computing applications.

我们提出了一种简单的算法，生成无限量的时间相关 Schrodinger 方程的两状态系统的精确解，此方法对量子计算应用中的量子比特控制特别有用。

可解析的驱动时变二能级量子系统

Analytically solvable driven time-dependent two-level quantum systems

Many real-world networks describe systems in which interactions decay with
the distance between nodes. Examples include systems constrained in real space
such as transportation and communication networks, as well as systems
constrained in abstract spaces such as multivariate biological or economic
datasets and models of social networks. These networks often display network
motifs: subgraphs that recur in the network much more often than in randomized
networks. To understand the origin of the network motifs in these networks, it
is important to study the subgraphs and network motifs that arise solely from
geometric constraints. To address this, we analyze geometric network models, in
which nodes are arranged on a lattice and edges are formed with a probability
that decays with the distance between nodes. We present analytical solutions
for the numbers of all 3 and 4-node subgraphs, in both directed and
non-directed geometric networks. We also analyze geometric networks with
arbitrary degree sequences, and models with a field that biases for directed
edges in one direction. Scaling rules for scaling of subgraph numbers with
system size, lattice dimension and interaction range are given. Several
invariant measures are found, such as the ratio of feedback and feed-forward
loops, which do not depend on system size, dimension or connectivity function.
We find that network motifs in many real-world networks, including social
networks and neuronal networks, are not captured solely by these geometric
models. This is in line with recent evidence that biological network motifs
were selected as basic circuit elements with defined information-processing
functions.

分析几何网络模型，提出了节点在点阵上排列的概念，并对 3 和 4 节点子图进行分析，发现许多真实世界的网络模式不仅仅是受到几何模型的影响，而是被选择为具有定义信息处理功能的基本电路元件。