On Maximum Complexity in Holography Shahrokh Parvizi ∗ and Mojtaba Shahbazi † Department of Physics, School of Sciences, Tarbiat Modares University, P.O.Box 14155-4838, Tehran, Iran July 13, 2021 Abstract In the quantum circuit, it is believed that complexity itself reaches a maximum of order exponential in the number of q-bits or equivalently exponential in entropy of the black hole. However, the current holographic proposals do not meet this criterion. The holographic proposals find the complexity of the very late times to be infinite, which makes it to be considered a non-physical quantity, while in the quantum circuit, it is expected that complexity meets within a finite time its maximum value. These points are required to be altered in holographic proposals of complexity. This letter introduces a new holographic proposal that meets all these criteria and consolidates the Lloyd bound. 1 Introduction There are some paradoxes in the combination of quantum mechanics and the general theory of relativity, mainly when one focuses on the physics of black holes by quantum mechan- ics' eye [1]. In this manner, the study of black hole physics may provide some meteoric arXiv:2103.05367v2 [hep-th] 11 Jul 2021 way to a successful quantum gravity theory. Recently Susskind's proposal based on the AdS/CFT correspondence made a big jump considering the interior of black holes [3]. By this conjecture, the volume of Einstein-Rosen bridges (ERB) corresponds to the complexity of conformal field theory on the boundary of an AdS space, which is coined as Complex- ity=Volume (CV) proposal. Complexity is the minimum number of quantum gates required to turn a reference state into a target state. Later, Brown et al. [4,5] suggested another holographic proposal dubbed as Complex- ity=Action (CA) proposal. It states that the action of bulk theory in Wheeler-DeWitt ∗[email protected] †[email protected] 1 patch, which is the domain of all space-like trajectories anchored to either side of the Penrose diagram is dual to the complexity of the boundary theory. It can be shown that in any quantum circuit, there is an upper bound of eK for the complexity, where K is the number of q-bits [3]. Because K number of q-bits is proportional S to the entropy S, we put K = S. In this way, the maximum complexity is Cmax = e [3,2]. Furthermore, the complexity growth rate is bounded from above by the mass of the system known as the Lloyd bound [7]. It has been revealed that some theories do not respect the Lloyd bound by CV and some by CA proposal in the full-time behavior [8], in Einstein-Maxwell-Dilaton theories [9, 10], in Lifshitz hyper scaling violating exponent space [11], in local quantum quench [12] and an anisotropic solution of Einstein-Dilaton-Axion theory [13]. This point leads to new proposals [9, 14] and modifications [15, 16, 17]. Nonetheless, none of these cases could maintain the criterion of maximum complexity because of the fact that they find the complexity for the late times infinite. Indeed, max- imum complexity arises at times of order exponential in entropy. Besides, the bulk side in AdS/CFT can not afford the quantum effects at late times regime [19]. The point is that by Poincar´erecurrence theorem, every unitary evolution by finite entropy will return to its initial state at late but finite times1. On the bulk side, correlation functions at late times are to be returned to a lower bound to respect the Poincar´erecurrence; however, one can not recover Poincar´erecurrence in the presence of black holes [19]. On its face, the maximum complexity that is reached at late times (the classical Poincar´erecurrence time) is a quantum effect and can not be recovered on the bulk side. Nonetheless, Maldacena's proposal in [19] by some averaging pursues recovering the recurrences at classical Poincare recurrence time. Then it makes sense to consider maximum complexity in holography. In this paper, we will provide a new holographic proposal that can fulfill the maximum complexity condition, and we will see that this new proposal behaves the Lloyd condition well, particularly for the large masses. Our proposal, which we call "Hyperbolic Proposal" could be seen either as an averaged complexity. In Sachdev-Ye-Kitaev (SYK) models that are examples of chaotic systems, the hyperbolic proposal bears a resemblance to averaged complexity. The organization of the paper is as follows: In the next section, the expected features of holographic complexity candidates are reviewed. Our new proposal is introduced in section 3, and we discuss its behavior. In section4, the relation between the hyperbolic proposal and averaged complexity is drawn, and we conclude in the final section. 1Non-unitary evolutions with finite entropy are not required to respect Poincar´erecurrence but it could be managed to have the recurrences by nonperturbative dynamical effects [20]. 2 2 Review on Expected Features of Holographic Com- plexity It is known that the computational complexity represents four expected features: linear growth, a particular behavior under perturbation, a maximum value of complexity, and the Lloyd bound [2, 22]. As a consequence, it is expected that the holographic picture of complexity embraces these features. One may consider the holographic complexity as a function like C(λ), where λ is a dynamical variable in the black hole and C a suitable function. In the case of well-known CV and CA proposals, λ is respectively the volume of the Einstein-Rosen bridge and the action in Wheeler-DeWitt patch, and C is simply a linear function with constant coefficients. First of all, we expect the holographic complexity C(λ) shows a linear growth in time and, secondly, behaves under perturbation as its dual in quantum circuits. Although it is expected that complexity reaches its maximum in a finite time, the current holographic proposals lack this condition. Let us discuss these behaviors in the following subsections and review the expected features in CV and CA proposals. 2.1 Linear growth In quantum circuits, it can be seen that restricted to a shorter time-scale when complexity is much smaller than its maximum, its growth is linear [3]: C / Kt where K is the number of q-bits. Although except at the beginning, it could be non-linear, it is believed that complexity grows all the way to its maximum linearly [22]. Let us start with the CV proposal and consider a black hole geometry of the following form: dr2 ds2 = −f(r)dt2 + + r2dΩ2 (1) f(r) D−2 µ f(r) = 1 + r2 − (2) rD−3 where for D > 3, 16πG M µ = N (3) (D − 2)!D−2 in which !D−2 is the volume of D − 2 dimensional sphere. The volume growth rate of ERB which is defined for a two-sided black hole is given by [3]: dV = ! rD−2pf(r) dt D−2 3 By induced metric for a volume in AdS-Schwazschild, the volume is written as [3]: s Z r02 V = r2 −f + dtdΩ (4) f 0 dr where r = dt . It could be shown that for finite times Eq. (4) turns into [3]: V / ST jtL + tRj (5) where S and T are entropy and temperature of the black hole, respectively, and tL and tR are times for the left and right side of the black hole. With C / V , Eq. (5) indicates linear growth of the holographic complexity. In the CA proposal, the computational complexity is proportional to the action of the theory in WDW patch, C / S. Due to the fact that the action is computed in a bounded manifold, the Gibbons-Hawking-York term should be added to the action of the theory: 1 Z p S = Stheory + hK 8πG @M where h is the induced metric on the boundary of the patch and K is extrinsic curvature constructed by the induced metric. At late but finite times, the complexity growth rate for a D dimensional AdS-Schwarzschild is given by [5]: D−1 dS ΩD−2rh = − 2 = 2ST (6) dt 8πGLAdS This shows the linear behavior as expected. The expressions (5) and (6) show that the current proposals of holographic complexity describe the linear growth of complexity. 2.2 Perturbation In [6], Susskind and Maldacena showed that by holography a pair of entangled particles is dual to an ERB or in short ER = EP R. Then a perturbation in gravity leads to a perturbation in the pair particles. Another test to check is on the perturbed thermo- field double (TFD) states which are dual of the geometries with shock waves [3, 21]. In other words, a perturbation in the evolution of black hole should hold in the correspondence between gravity and quantum circuit. Small perturbations in TFD state could be conducted by precursors as: j (t)i = W (t)j i where W (t) = U yWU with W close to one and U = e−iH . The quantum circuit of the precursors could be seen in Fig.1. The complexity of the perturbed TFD state is read as [3]: C / K(tf − 2npt∗) (7) 4 Figure 1: A circuit of precursor. The horizontal lines are q-bits and the circles are quantum gates. Figure 2: A Penrose diagram for a two-sided black hole with a shock wave at time tw. This is supposed to be dual to a precursor. where tf is the total time of perturbation, np is the number of stages of perturbation and t∗ is the scrambling time. The dual picture is the geometries with np number of shock waves as in Fig.2[3].