# Weighting gates in circuit complexity and holography

## TL;DR

This paper explores how assigning different weights to quantum gates affects the complexity of quantum circuits, revealing how penalties influence divergence behavior and connecting these findings to holographic complexity predictions.

## Contribution

It introduces a detailed analysis of gate weighting effects on circuit complexity in field theories and relates these effects to holographic complexity models, highlighting the impact on divergence and scaling.

## Key findings

- Higher gate costs increase complexity.
- Penalties modify UV divergence, removing logarithmic factors.
- Relation established between weighted complexity and holographic predictions.

## Abstract

Motivated by recent studies of quantum computational complexity in quantum field theory and holography, we discuss how weighting certain classes of gates building up a quantum circuit more heavily than others does affect the complexity. Utilizing Nielsen's geometric approach to circuit complexity, we investigate the effects for a regulated field theory for which the optimal circuit is a representation of $GL(N,\mathbb{R})$. More precisely, we work out how a uniformly chosen weighting factor acting on the entangling gates affects the complexity and, particularly, its divergent behavior. We show that assigning a higher cost to the entangling gates increases the complexity. Employing the penalized and the unpenalized complexities for the $\mathcal{F}_{\kappa=2}$ cost, we further find an interesting relation between the latter and the one based on the unpenalized $\mathcal{F}_{\kappa=1}$ cost. In addition, we exhibit how imposing such penalties modifies the leading order UV divergence in the complexity. We show that appropriately tuning the gate weighting eliminates the additional logarithmic factor, thus, resulting in a simple power law scaling. We also compare the circuit complexity with holographic predictions, specifically, based on the complexity=action conjecture, and relate the weighting factor to certain bulk quantities. Finally, we comment on certain expectations concerning the role of gate penalties in defining complexity in field theory and also speculate on possible implications for holography.