# Circuit Complexity across a Topological Phase Transition

## TL;DR

This paper investigates how Nielsen's circuit complexity can detect topological phase transitions in a Kitaev chain, revealing non-analytical behaviors at critical points and classifying phases based on complexity properties.

## Contribution

It demonstrates that circuit complexity captures topological phase transitions and the locality of optimal Hamiltonians, extending to long-range and higher-dimensional systems.

## Key findings

- Complexity exhibits non-analytical behavior at critical points.
- Locality of optimal Hamiltonian depends on phase membership.
- Results extend to long-range and higher-dimensional models.

## Abstract

We use Nielsen's geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and non-equilibrium steady states of the Kitaev model exhibit non-analytical behaviors at the critical points, and thus can be used to detect both {\it equilibrium} and {\it dynamical} topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen's circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and discuss generalizations to higher dimensions. Our result opens up a new avenue for using circuit complexity as a novel tool to understand quantum many-body systems.