# Holographic Complexity and Charged Scalar Fields

## TL;DR

This paper develops a time-dependent measure of quantum complexity for charged scalar fields in a holographic setting, showing linear growth over time and adherence to the Lloyd bound for small charges.

## Contribution

It introduces a new holographic complexity measure for charged scalar fields in AdS black hole backgrounds, analyzing its time evolution and bounds.

## Key findings

- Complexity grows linearly over time for large intervals.
- For small charges, the growth rate respects the Lloyd bound.
- The framework applies to coupled conformal scalar field theories in holography.

## Abstract

We construct a time-dependent expression of the computational complexity of a quantum system which consists of two conformal complex scalar field theories in d dimensions coupled to constant electric potentials and defined on the boundaries of a charged AdS black hole in (d+1) dimensions. Using a suitable choice of the reference state, Hamiltonian gates and the metric on the manifold of unitaries, we find that the complexity grows linearly for a relatively large interval of time. We also remark that for scalar fields with very small charges the rate of variation of the complexity cannot exceed a maximum value known as the Lloyd bound.