# Principles and symmetries of complexity in quantum field theory

## TL;DR

This paper develops a geometric framework for continuous quantum complexity using Finsler geometry, revealing symmetries and proposing a new interpretation of quantum evolution as cost minimization.

## Contribution

It introduces a Finsler geometric approach to quantum complexity in continuous systems, highlighting bi-invariance and deriving unique complexity measures for SU(n) operators.

## Key findings

- Finsler metric naturally emerges in continuous quantum complexity
- Complexity of SU(n) operators is a length of geodesics in Finsler geometry
- Proposes quantum state evolution as a process of minimizing computational cost

## 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 Schr\"{o}dinger's equation in isolated systems - the quantum state evolves by the process of minimizing "computational cost."