Weighting gates in circuit complexity and holography. (arXiv:1903.06156v1 [hep-th])

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.

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