Principles and symmetries of complexity in quantum field theory

TL;DR

This work constructs a geometric theory of operator complexity for continuous quantum systems by deriving a bi-invariant Finsler metric on SU(n) from four axioms (G1-G4) and essential QFT symmetries. The complexity of an operator is the minimal geodesic length, with a universal form F(c, dot c) = $\lambda$ Tr sqrt($H H^{\dagger}$) and geodesics generated by constant generators, linking complexity to Schrödinger evolution via a Minimal Cost Principle. It clarifies distinctions and connections to discrete, k-local qubit models, showing adjoint invariance and path-reversal symmetry, and proposes a framework that unifies operator complexity with CPT/unitary symmetries while offering a route toward holographic complexity concepts. The results provide a rigorous, symmetry-driven foundation for complexity in QFT and illuminate how continuous notions of cost encode quantum dynamics."

Abstract

Based on general and minimal properties of the {\it discrete} circuit complexity, we define the complexity in {\it continuous} systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU($n$) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems. We clarify why the bi-invariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in $k$-local right-invariant metric can also appear in our framework. Based on the bi-invariance of our formalism, we propose a new interpretation for the Schrödinger's equation in isolated systems - the quantum state evolves by the process of minimizing "computational cost."

Figures (7)

• Figure 1: The logical flows of this paper. By answering three basic questions at the far left side we show that the complexity geometry is determined by a unique bi-invariant Finsler geometry. As an application of our formalism, we show that the Schrödinger's equation for isolated systems can be obtained from a "minimal cost principle". Although the bi-invariant geometry looks very different from the right-invariant $k$-local Riemannian geometry proposed by Ref. Brown:2017jil, we will show that for all $k$-local operators two theory will give equivalent results.
• Figure 2: Schematic diagram for the complexity of the Cartesian product and parallel decomposition rule. As two operators $\hat{x}_1$ and $\hat{x}_2$ are simulated independently, the minimally required gates for $(\hat{x}_1,\hat{x}_2)$ is the sum of the minimally required gates for $\hat{x}_1$ and $\hat{x}_2$. Thus, we have $\mathcal{C}((\hat{x}_1,\hat{x}_2))=\mathcal{C}((\hat{x}_1,\hat{\mathbb{I}}_2))+\mathcal{C}((\hat{\mathbb{I}}_1,\hat{x}_2))$.
• Figure 3: A curve $c(s)$ connects the identity and a particular operator $\hat{O}$ with the endpoints $c(0)=\hat{\mathbb{I}}$ and $c(1)=\hat{O}$. This curve can be approximated by a discrete form. Every endpoint is also an operator, which is labeled by $\hat{O}_n$
• Figure 4: Schematic diagram for two different generators in quantum circuits. To obtain the some target operator $\hat{O}$ from the quantum circuit $\phi_0$, we have two different ways to add new circuits.
• Figure 5: Equivalences between i) the unitary-invariance of QFT, ii) the independence of the Finsler metric on the left/right generators, and iii) the adjoint invariance of the complexity