Circuit Complexity for Coherent States

TL;DR

This work extends circuit-complexity analysis to coherent states in a free scalar field, using Nielsen’s geometric approach and contrasting it with the Fubini-Study method. By embedding the problem in an extended gate set that includes shifts, the authors derive a rich geometric structure (notably an $\mathbb{R}^{N} \rtimes GL(N,\mathbb{R})$ group) and obtain analytic and numerical results for simple geodesics, plus perturbative results for small excitations, showing that intermediate states generally become entangled even when endpoints do not. They examine multiple cost functions ($F_2$, $F_1$, $\kappa=2$, Schatten$\,p$) and demonstrate that, while the finite increase over the vacuum is UV-finite in many cases, the computed complexities differ across Nielsen vs FS formalisms, with partial agreement under specific scale choices. The QFT extension via lattice regularization reveals mode-decoupling at leading order and additive per-mode contributions to complexity increases, offering insights relevant to holographic complexity and potential extensions to time evolution and momentum-coherent states. Overall, the paper clarifies how coherent-state preparation probes the structure of circuit complexity in QFT and highlights both commonalities and distinctions between disparate complexity frameworks.

Abstract

We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen's geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen's approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.

Figures (10)

• 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 all of the intermediate states ${\left\vert{\psi_i}\right\rangle}$ that are produced after every step.
• Figure 2: Comparing the numerical solutions to analytic solutions for the simple geodesics (\ref{['eq:solution']}). The top two graphs show the geodesic ending at $y_{+1}=0.211,\ y_{-1}=1.211,\ u_{+1}=1.690,\ u_{-1}=0$, while the bottom two graphs show the geodesic ending at $y_{+1}=-0.790,\ y_{-1}=0.211,\ u_{+1}=0,\ u_{-1}=1.690$. These values were chosen to produce simple values for $\Lambda_+$ and $\Lambda_-$, i.e.,$\Lambda_+=\Lambda_-=1$. The subscripts ''n'' and ''a'' are used to indicate the numerical and analytical solutions, respectively.
• Figure 3: Lengths of a family of geodesics ($k$) connecting to two target states, in which a single mode is excited, for different final values of the $z$ angle. The red upper triangles represent geodesics reaching the state with $y_{+1}=0.1,\ y_{-1}=1.1,\ \Lambda_{+}= 0,\ \Lambda_{-}=6.008$. The blue lower triangles are for $y_{+1}=0.1,\ y_{-1}=1.1,\ \Lambda_{+}=1.105,\ \Lambda_{-}=0$. In both cases, the minimum value arises at $z=0$, i.e., the optimal geodesic corresponds to one of the simple geodesics found in the previous section.
• Figure 4: Example of geodesics preparing a target state with both $a_\pm$ nonvanishing. We compare the optimal geodesic and the ''simpler'' geodesic with $z_1=0$. In this example, both geodesics have the final boundary conditions: $y_{+1}=0.1,\ y_{-1}=1.1, \ \Lambda_{+}=1.105,\ \Lambda_{-}=6.008$. For the optimal geodesic, we also have $x_0=z_0=-0.0976\pi, x_1=0, z_1=-0.0167\pi$, while for the ''simpler'' one, $x_0=z_0=-0.0749\pi, x_1=0= z_1$. Note that $y_{\pm}$ essentially coincide in both geodesics, as shown in the far left panel.
• Figure 5: A comparison of the lengths of the optimal geodesic and the "simpler" geodesic with $z_1=0$. $\Delta k= k_{sim} -k_{opt}$ and $k_{opt}$ are shown as functions of $\Lambda_{-}(s=1)=-\frac{m\omega_+}{x_0\omega_{\textrm{\tiny R}}^2}\,a_-$. This example is characterized by the boundary conditions $y_{+1}=0.1,\ y_{-1}=1.1$ and $\ \Lambda_{+1}=1.105$, while $\Lambda_{-}(s=1)$ varies from 0 to 12.017. We note that while $\Delta k$ grows as $|a_-|$ increases, it represents at most an increase of $~0.2\%$ over $k_{opt}$ for the geodesics shown here.