Weighting gates in circuit complexity and holography

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