Abstract
It is conjectured that the average energy provides an upper bound on the rate at which the complexity of a holographic boundary state grows. In this paper, we perturb a holographic CFT by a relevant operator with a time-dependent coupling, and study the complexity of the time-dependent state using the complexity equals action and the complexity equals volume conjectures. We find that the rate of complexification according to both of these conjectures has UV divergences, whereas the instantaneous energy is UV finite. This implies that neither the complexity equals action nor complexity equals volume conjecture is consistent with the conjectured bound on the rate of complexification.
Highlights
The complexity of a quantum state is defined as the minimum number of gates that map a reference state to that quantum state [1,2,3]. It was conjectured in [1,2] that the growth of the complexity of a boundary CFT state as a function of time is holographically dual to the stretching of the interior of a black hole in the bulk
Known as the complexity equals action (CA) conjecture, relates the holographic complexity with the on-shell gravitational action of a certain bulk region [3,4]. This region, called the Wheeler-DeWitt (WDW) patch, is defined as the domain of dependence of a bulk Cauchy surface. This conjecture states that the complexity of a CFT state at time t is related to the onshell action of the WDW patch corresponding to time t, AðtÞ, as [3,4]
Our goal is to present another example where the growth of the complexity computed using either the CA or the complexity equals volume (CV) conjecture does not respect the Lloyd bound
Summary
The complexity of a quantum state is defined as the minimum number of gates that map a reference state to that quantum state [1,2,3]. The UV divergences in the on-shell action of the WDW patch and in the volume of a Cauchy slice only depend on the asymptotic geometry near the boundary [20,21]. This means that it is sufficient to solve the bulk equations perturbatively in the radial coordinate. The response of the one-point function of the boundary stress tensor to the time-dependent perturbation is studied using the AdS-CFT correspondence in [24,25] This requires variation of the on-shell renormalized action of the bulk theory with respect to the metric of the boundary. All we need to know for the violation of the Lloyd bound is the fact that the energy is a UV finite quantity
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