Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.

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克雷洛夫和尼尔森复杂度之间的关系

克雷洛夫复杂性和尼尔森复杂性是量化量子演化复杂性的成功方法，人们一直在积极追求这些方法，而这两个研究领域之间没有太多联系。这两个量分别是由量子混沌和量子计算驱动的，而相关数学与矩阵对角化算法和曲线流形上的测地流一样不同。我们证明，尽管存在这些差异，但这两个量之间仍然存在关系。也就是说，状态演化的克雷洛夫复杂度的时间平均值可以表示为某个矩阵的迹，该矩阵还通过适应克雷洛夫基础的特定定制惩罚表来控制尼尔森复杂度的上限。