Aspects of the first law of complexity

Abstract

We investigate the first law of complexity proposed in [1], i.e., the variation of complexity when the target state is perturbed, in more detail. Based on Nielsen's geometric approach to quantum circuit complexity, we find the variation only depends on the end of the optimal circuit. We apply the first law to gain new insights into the quantum circuits and complexity models underlying holographic complexity. In particular, we examine the variation of the holographic complexity for both the complexity=action and complexity=volume conjectures in perturbing the AdS vacuum with coherent state excitations of a free scalar field. We also examine the variations of circuit complexity produced by the same excitations for the free scalar field theory in a fixed AdS background. In this case, our work extends the existing treatment of Gaussian coherent states to properly include the time dependence of the complexity variation. We comment on the similarities and differences of the holographic and QFT results.

Circuit complexity for coherent states

“…However, the F 1 cost function does not exhibit this behaviour. We plan to return to this topic and make a detailed comparison between our results for the complexity of QFT coherent states and holographic complexity in [74].…”

Classical limit of large N gauge theories with conformal symmetry

“…This is the liner approximation of the Einstein gravity (at least if α i → 0 in the large N limit). Here, we describe the coherent states for this approximation, which have been considered in [29,30].…”

Time dependence of complexity for Lovelock black holes

“…Here ct is an arbitrary length scale, Θ is the scalar expansion defined by Θ = ∇ µ k µ andΘ = ∇ µ sk µ . Though the inclusion of this counter term is not necessary for the action principle, it plays an indispensable role in shockwave geometries [18,19,20,21] as well as in other time dependent geometries [22]. However, for static black holes it does not have any time dependence and hence is irrelevant to current paper.…”

“…In the AdS/CFT correspondence, there are two popular proposals for complexity, usually dubbed by "Complexity=Volume" (CV) duality [7,8] and "Complex-ity=Action" (CA) duality [9,10]. The two conjectures have been extensively studied in literature [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. However, there are some drawbacks for the two proposals as well.…”