We investigate the first law of complexity proposed in Bernamonti et al (2019 Phys. Rev. Lett. 123 081601), i.e., the variation of complexity when the target state is perturbed, in more detail. Based on Nielsen’s geometric approach to quantum circuit complexity, we find the variation only depends on the end of the optimal circuit. We apply the first law to gain new insights into the quantum circuits and complexity models underlying holographic complexity. In particular, we examine the variation of the holographic complexity for both the complexity = action and complexity = volume conjectures in perturbing the AdS vacuum with coherent state excitations of a free scalar field. We also examine the variations of circuit complexity produced by the same excitations for the free scalar field theory in a fixed AdS background. In this case, our work extends the existing treatment of Gaussian coherent states to properly include the time dependence of the complexity va...

中文翻译：

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复杂性第一定律的方面

我们更详细地研究了Bernamonti等人（2019 Phys.Rev.Lett.123081601）中提出的第一条复杂性定律，即当目标状态受到扰动时复杂性的变化。基于尼尔森针对量子电路复杂性的几何方法，我们发现变化仅取决于最佳电路的末端。我们应用第一定律来获得有关全息复杂性的量子电路和复杂性模型的新见解。特别是，我们在自由标量场的相干态激发下扰动AdS真空时，针对复杂度=动作和复杂度=体积猜想，研究了全息复杂度的变化。我们还研究了在固定AdS背景下，相同的激励对自由标量场理论产生的电路复杂性的变化。在这种情况下，