Weighting gates in circuit complexity and holography

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 affects 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 penalized and unpenalized complexities for the $\mathcal{F}_{\kappa=2}$ cost, we further find an interesting relation between the latter and that 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.