Circuit complexity in quantum field theory

TL;DR

This work develops a Nielsen-geometric framework to define and compute circuit complexity for Gaussian states in a free scalar quantum field theory, starting from a simple two-oscillator warm-up and extending to a lattice of oscillators. The complexity is captured as geodesic length in a GL(2,R) circuit space, which simplifies in the normal-mode basis where the ground state becomes a product of Gaussians and the optimal circuit amounts to linearly amplifying each normal mode. The lattice generalization yields an explicit leading-divergence form that can mirror holographic CA/CV behavior depending on the chosen cost function, and the introduction of penalty factors demonstrates nontrivial, locality-aware deformations of the optimal circuit, including a new segmented geodesic that becomes favorable at large penalties. Overall, the paper connects quantum-field-theoretic circuit complexity to holographic ideas, proposes concrete computational schemes, and outlines rich avenues for future exploration including cMERA connections and extensions to interacting theories.

Abstract

Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.

Figures (8)

• Figure 1: Complexity=volume (CV, left) and complexity=action (CA, right) for the eternal AdS black hole dual to the thermofield double state \ref{['eq:TFD']}. In the left panel, the blue curve represents the maximal spacelike surfaces that connects the specified time slices on the left and right boundaries. In the right image, the shaded region is the corresponding WDW patch.
• Figure 2: Sketch of the one-parameter family of geodesics. The vertical axis is $\tau$, the horizontal plane is described by the radius $\rho$ and the azimuthal angle $\theta$, and the $y$ direction is suppressed. The circuits which produce the transformation from $A_\textrm{\tiny R}$ to $A_\textrm{\tiny T}$ are described by geodesics running from the origin to the blue spiral at $\theta+\tau=\theta_1+\tau_1$ and $\rho=\rho_1$ (shown here for the special case $\theta_1+\tau_1=\pi$, which appears in eq. (\ref{['fin2']}) below). The black curves represent (non-minimal) geodesics within the one-parameter family of solutions with different values of $\theta_0$. The minimum geodesic corresponds to the green line in the $\tau=0$ plane with $\Delta\theta=0$ ( i.e., $\theta_0=\theta_1$), whose length is given by eq. \ref{['solver4b']}.
• Figure 3: (Left:) Plot of $\hat{k}$ (red), $\hat{k}_0$ (blue), and $\hat{k}_s$ (green, $y_1\in\{0,20,100,200\}$) as given in eqs. \ref{['eq:vs']} and \ref{['eq:kStraight']} as functions of the penalty factor $\frak{a}$, with $\epsilon=10^{-10}$. Clearly, $\bar{k}$ represents a shorter geodesic than the straight-line circuit with $\bar{k}_0$. This again indicates the existence of short-cuts outside of the normal-mode subspace in the penalized geometry. For $\bar{k}_s$, the opacity reflects the value of $y_1$, with $y_1=0$ the lowest/darkest curve running parallel to $\hat{k}$ and $y_1=200$ the highest/faintest. (Right:) Plot of $\left(\bar{k}_s-\bar{k}\right)/\bar{k}$ as a function of $\frak{a}$ for $\epsilon=10^{-10}$, where the shading runs through the same range of $y_1$ as in the left plot (from $y_1=0$ at the bottom to $y_1=200$ at the top). Though it is not clear at this scale, the lower-most curve, $y_1=0$, follows the same basic shape as the others, peaking at $\frak{a}\sim0.1$ and then slowly approaching to zero as $a\rightarrow\infty$. The curves never become negative, indicating that $\bar{k}_s>\bar{k}$ for all $\frak{a}>1$.
• Figure 4: Plot of $\bar{k}_s-\bar{k}$ as a function of $\frak{a}$ for $\epsilon=10^{-10}$, where the shading runs through the same range of $y_1$ as in figure \ref{['fig:kCompare']} (from $y_1=0$ at the bottom to $y_1=200$ at the top). In all cases, the difference eventually approaches $1/2$, consistent with eq. \ref{['eq:overhead']}. For example, when $\frak{a}=1000$, we have $\bar{k}_s\simeq791.848$ and $792.052$ for $y_1=0$ and $200$, respectively, while $\hat{k}\simeq 791.348$.
• Figure 5: (Left) Plot of $x(s)$ (blue) and $\rho(s)$ (red) for the parameter values set by \ref{['eq:kEps']} and \ref{['eq:params']}, with $\epsilon=10^{-16}$. (Right) Plot of $\dot x(s)$ (blue) and $\dot\rho(s)$ (red) for the same. In both plots, the dashed curves correspond to $\frak{a}=2$, while for the solid curves we have set $\frak{a}=50$. One sees that the amount of "time" for which the circuit remains on the constant $x=\pi/4$ segment is inversely proportional to the strength of the penalty factor. For illustrative purposes, we have normalized $\rho|_{s=1}$ to 2 in the left plot, and normalized both $\dot x|_{s=1}$ and $\dot\rho|_{s=1}$ to 1 in the right.