Aspects of The First Law of Complexity

TL;DR

The authors formulate and test a first-law-like relation for circuit complexity within Nielsen's geometric framework, using coherent-state perturbations in AdS/CFT to connect quantum-circuit notions with holographic complexity. They develop a detailed holographic analysis of complexity variations under small scalar perturbations for both CA and CV, including analytic and numerical results, and contrast these with a parallel QFT treatment of a free scalar in a fixed AdS background. The work highlights both the shared second-order dependence on perturbation amplitudes and the distinct time dependence and off-diagonal structures that arise in holographic settings, suggesting a richer gate-structure in holography than in the Gaussian QFT model. Overall, the paper strengthens the bridge between holographic observables and quantum-circuit notions of complexity, while outlining clear avenues for broader applications and deeper understanding of holographic gates and cost functions.

Abstract

We investigate the first law of complexity proposed in arXiv:1903.04511, 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 variation. We comment on the similarities and differences of the holographic and QFT results.

Figures (15)

• Figure 1: A general quantum circuit where ${\left\vert{\psi_\textrm{\tiny T}}\right\rangle}$ is prepared beginning with ${\left\vert{\psi_\textrm{\tiny R}}\right\rangle}$ and applying a sequence of elementary unitaries $g_{i}$. We also indicate the intermediate states that are produced after every step, i.e.,${\left\vert{\psi_k}\right\rangle}=g_{i_\textrm{\tiny k}}g_{i_\textrm{\tiny k--1}}\cdots g_{i_\textrm{\tiny 2}}\,g_{i_\textrm{\tiny 1}}{\left\vert{\psi_\textrm{\tiny R}}\right\rangle}$.
• Figure 2: The variation of the Nielsen circuit due to a perturbation ${\left\vert{\Psi_{\textrm{\tiny T}} +\delta\Psi}\right\rangle}$ of the target state ${\left\vert{\Psi_{\textrm{\tiny T}}}\right\rangle}$.
• Figure 3: Representation of the WDW patch. The WDW patch is bounded by the future and past null surfaces $t_\pm(\rho)$ (thick blue lines) joining at $t_\Sigma$ on the AdS conformal boundary (grey line). $k_{\mu} dx^\mu$ is the outward directed normal one-form to the null WDW boundary. The regulated asymptotic AdS boundary (red line) cuts the WDW patch at $\rho =\pi/2 - \epsilon_\rho$, and has outward directed normal $n_\mu dx^\mu$. The $\rho =\pi/2 - \epsilon_\rho$ regulator surface and null hypersurfaces intersect at the null joint codimension-2 surfaces at $t_\pm(\pi/2 -\epsilon_\rho)$.
• Figure 4: The joint between the null surface and the time-like regulator surface. $k_\mu dx^\mu$ and $n_\mu dx^\mu$ are the outward directed normal one forms. $\hat{t}^{\mu} \partial_{\mu}$ is a unit vector in the tangent space to the boundary time-like surface and outward directed with respect to the boundary of this surface.
• Figure 5: Explicit results for $\mathcal{C}^\textrm{\tiny A}_{j,k}$ and $\mathcal{S}^\textrm{\tiny A}_{j,k}$ with fixed $k$ and $d=3=\Delta$.

Theorems & Definitions (1)

• Theorem 1