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Motivated by recent studies of quantum computational complexity in quantumfield theory and holography, we discuss how weighting certain classes of gatesbuilding up a quantum circuit more heavily than others does affect thecomplexity. Utilizing Nielsen's geometric approach to circuit complexity, weinvestigate the effects for a regulated field theory for which the optimalcircuit is a representation of $GL(N,\mathbb{R})$. More precisely, we work outhow a uniformly chosen weighting factor acting on the entangling gates affectsthe complexity and, particularly, its divergent behavior. We show thatassigning 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 betweenthe latter and the one based on the unpenalized $\mathcal{F}_{\kappa=1}$ cost.In addition, we exhibit how imposing such penalties modifies the leading orderUV divergence in the complexity. We show that appropriately tuning the gateweighting eliminates the additional logarithmic factor, thus, resulting in asimple power law scaling. We also compare the circuit complexity withholographic predictions, specifically, based on the complexity=actionconjecture, and relate the weighting factor to certain bulk quantities.Finally, we comment on certain expectations concerning the role of gatepenalties in defining complexity in field theory and also speculate on possibleimplications for holography.

Weighting gates in circuit complexity and holography