More on complexity of operators in quantum field theory

TL;DR

<p>The paper generalizes operator complexity in SU(n) quantum-field theories by relaxing the parallel-decomposition axiom and the smoothness assumption, yielding a Schatten p-norm cost with an optional penalty w. With adjoint invariance and CPT reversibility, the Finsler metric is fixed up to a constant, giving a Schatten-p form for the complexity of an operator O via its logarithm H-bar = ln O. The work shows consistency with right-invariant geometries and provides a concrete realization of linear-growth-to-saturation dynamics, with the saturation time tied to the topology and curvature of SU(n). It also analyzes geodesic deviation, chaos, and the complexity of precursors, highlighting that negative sectional curvature can arise on submanifolds within a bi-invariant framework. Overall, the results extend the complexity framework in QFT and offer a unifying view of operator complexity, geometric growth, and precursor dynamics.</p>

Abstract

Recently it has been shown that the complexity of SU($n$) operator is determined by the geodesic length in a bi-invariant Finsler geometry, which is constrained by some symmetries of quantum field theory. It is based on three axioms and one assumption regarding the complexity in continuous systems. By relaxing one axiom and an assumption, we find that the complexity formula is naturally generalized to the Schatten $p$-norm type. We also clarify the relation between our complexity and other works. First, we show that our results in a bi-invariant geometry are consistent with the ones in a right-invariant geometry such as $k$-local geometry. Here, a careful analysis of the sectional curvature is crucial. Second, we show that our complexity can concretely realize the conjectured pattern of the time-evolution of the complexity: the linear growth up to saturation time. The saturation time can be estimated by the relation between the topology and curvature of SU($n$) groups.

Figures (5)

• Figure 1: Schematic diagram for G3a in quantum circuits. Two operators $\hat{x}_1$ and $\hat{x}_2$ are independent so it is natural that $\mathcal{C}((\hat{x}_1,\hat{x}_2))=\mathcal{C}((\hat{x}_1,\hat{\mathbb{I}}_2))+\mathcal{C}((\hat{\mathbb{I}}_1,\hat{x}_2))$.
• Figure 2: A continuous curve $c(s)$ connects the identity ($c(0)=\hat{\mathbb{I}}$) and a particular operator $\hat{O}$ ($c(1)=\hat{O}$). It can be approximated by a discrete form, where every intermediate point ($\hat{O}_n$) is also an operator.
• Figure 3: The schematic figure to show that the left generator is not unique for a given right generator $H_r$. The two left generators can be connected as: $H_l^{(2)}=\hat{U}H_l^{(1)}\hat{U}^{-1}$ with $\hat{U}=\hat{U}_2^{-1}\hat{U}_1$.
• Figure 4: The conjectured schematic diagram for the complexity evolution of the operator $\exp(Hs)$, where $H$ is a constant generator in $\mathfrak{su}(n)$. The complexity first grows linearly when $s<s_c$ and reaches its maximum at the time $s=s_c\leq\mathcal{O}(\sqrt{n}) \propto \mathcal{O}(e^{d/2})$, where $d$ is proportional to the classical degree of freedom of the system i.e., the size of classical phase space. At a very large time $s = s_r \propto\mathcal{O}(\exp(e^{d}))$, the quantum recurrence occurs and the complexity goes down to zero.
• Figure 5: Left panel: geometric explanation of ${\hat{W}}(t)={\hat{U}}(t){\hat{W}}_0{\hat{U}}(-t)$ and distance $d(t)$ (relative complexity). Right penal: $c_0(s)$ and $c_t(s)$ are two geodesics connecting from $\hat{\mathbb{I}}$ to ${\hat{W}}_0$ and ${\hat{W}}(t)$ respectively. It is possible that two geodesics $c_0(s)$ and $c_t(s)$ have the same length even though ${\hat{W}}(t)\neq W_0$.