Holographic Interpretation of Relative State Complexity

We propose, within the AdS/CFT correspondence, a holographic interpretation of relative state complexity. The scenario we examine involves a large-$N$ CFT in a high-energy pure state coupled to a small auxiliary system of $M$ weakly-interacting degrees of freedom. We conjecture the relative state complexity of the auxiliary system is dual to an effective low-energy notion of a computational cost (decoding task) in the bulk, i.e. to the minimal depth of the quantum circuit required to decode the auxiliary CFT.


Introduction
The AdS/CFT correspondence relates a d-dimensional boundary conformal field theory (CFT) to a gravity theory in (d+1)-dimensional asymptotically Anti-de Sitter (AdS) spacetime. The duality points to a deep relation between the concepts of quantum computational complexity, quantum information, chaos, and gravity, all of which interplay in black holes. Quantum computational complexity (herein referred to as complexity) is usually defined as the minimum number of gates required to prepare a quantum state, assuming a universal gate set {g i }. The holographic complexity of a boundary CFT has been shown to be dual to (i) the volume of a maximally extended spacelike hypersurface (i.e. codimension-one submanifold) behind the black hole horizon (CV-duality), and (ii) the classical action of the Wheeler-DeWitt (WDW) patch (CA-duality). The CV-duality dictates the growth of holographic complexity [5,23] is gravitationally dual in the AdS bulk to the linear growth of the black hole interior, where the exponential upper bound of complexity corresponds to the exponentially long time for which the interior grows. 1 This duality suggests there is behind-the-horizon dynamics which goes on for time exponential in the number of degrees of freedom after the coarse-grained entropy is saturated. The CA-duality, on the other hand, associates a family of weakly coupled bulk degrees of freedom to the quantum state of the boundary CFT. As it was demonstrated in [15,21] the CA-duality is more fundamental, and hence can reproduce the CV-duality for AdS black holes of arbitrary masses and in arbitrary number of dimensions. 2 We can thus express, up to a length term, the volume growth of a hypersurface, anchored at some boundary time, in terms of the growth of the classical action [21]. Another notion which we will examine in this paper is that of relative state complexity, defined as the minimum number of gates required to prepare a target state |ψ from a simple reference state |ψ 0 , assuming a universal gate set {g i }. Usually, for a two-sided AdS black hole the reference state is the thermofield-double (TFD) state, while for a one-sided AdS black hole it is the ground state. In fact, as we will assume throughout the paper for the case of a one-sided AdS black hole, the target state will simply be the time-evolved reference state |ψ = U |ψ 0 , where U denotes a unitary transformation. Note that the entropic behavior of complexity [20] suggests the relative state complexity for any |ψ and |ψ increases linearly with the number of degrees of freedom. Our goal in this paper is to propose, within AdS/CFT, a possible interpretation of relative state complexity as a decoding task in the bulk. Throughout the paper we will assume the idealized scenario in which the CFT evolution is described by a random quantum circuit acting on a large but finite N . 3 Moreover, we will assume the random quantum circuit has a discrete time-step evolution ∆τ , dictated by a universal gate set of two-local gates. 4 We investigate the evolution of relative state complexity of a small auxiliary system of M weakly interacting degrees of freedom which is coupled to a large-N CFT in a high-energy pure state (gravitationally dual to one-sided AdS black hole). We suggest the relative state complexity can be related to an effective low-energy notion of computational cost in the bulk. In particular, we interpret the relative state complexity of the auxiliary system as a decoding task in the bulk, reminiscent of the Harlow-Hayden computational cost argument [19] in the context of decoding subfactors of the Hilbert space of the Hawking radiation in the firewall proposal.

Complexity and Chaos
In this Section we examine how complexity and chaos develop in large-N random quantum circuits within AdS/CFT. We then put forward, employing Nielsen's approach, a geometrical interpretation of complexity and chaos, where the computational cost is given in terms of "distance" on a unitary manifold [22]. Later, we demonstrate the same geometrical interpretation naturally arises in the large-N limit of the Sachdev-Ye-Kitaev (SYK) model.

Complexity
Complexity has quasi-periodic behavior. For a generic large-N random quantum circuit, complexity is low for small t. Then, at the scrambling time, controlled by the Lyapunov 2 Using the classical action associated with the WDW patch in AdS eliminates the ambiguities related to the size of the horizon relative to the AdS radius.
3 Throughout the paper we will work under the assumption that the black hole is represented as a collection of N qubits, where SBH = A 4G N = N . exponent (2.6), and bounded from above by the Lloyd's bound (2.3), complexity grows exponentially. Having the Lyapunov exponent dominate the dynamics at the scrambling time indicates the presence of chaos in the CFT. Later on, however, for large t, the exponential growth of complexity is saturated. Thus, after the scrambling time, complexity continues to increase but now linearly in the number of degrees of freedom (2.16) for time t ∼ e N (classical recurrence time) at which point the upper bound C max = poly(N )e N is reached. Complexity then remains at its maximum value for a quantum recurrence time t qr ∼ e e N (doubly exponential in the entropy) and then it begins to decrease. For a system of N degrees of freedom at temperature T , the rate of growth of complexity at the scrambling time is Although the scrambling complexity is nowhere near the upper bound C * C max , it is still substantial for N 1. After the scrambling time complexity continues to grow but now linearly in N [18] where both early-and late-time growth (2.1) and (2.2), respectively, are restricted by the Lloyd's bound [17] The sharp transition in the dynamics at the scrambling time is well-motivated on both sides of the duality. In particular, if this exponential growth is saturated before the scrambling time, then this would indicate some yet unknown interior dynamics for AdS black holes which allows for faster information processing. On the other hand, exponential growth of complexity beyond the scrambling time would violate (2.3), see Refs. [16,22]. Apparently, the scrambling time is of particular interest when studying large-N chaotic systems. It is given as Having a two-local universal gate set, we can define the scrambling time as the time for a reduced density matrix to become approximately thermal [15]. Another definition, suitable for strongly-coupled large-N theories (dual to AdS black holes), is the time for all degrees of freedom to indirectly interact. From the bulk/boundary equality of the Hilbert spaces, we see the scrambling time is of particular importance for both AdS black holes, and high-temperature boundary CFTs since it indicates the presence of chaos.

Chaos
Chaos quantifies the sensitivity of a system to changes in the initial conditions. The chaotic behavior of a strongly-coupled large-N CFT manifests in the AdS bulk as fast scrambling (2.4). A commonly used way to probe chaos involves the use of out-of-time-order correlators, where for a strongly-coupled large-N quantum system at some fixed temperature β Here, λ L is the Lyapunov exponent which is bounded from above as W and V are simple Hermitian operators, where and H is a local Hamiltonian. At the scrambling time, due to chaos, the out-of-time-order correlator (2.5), up to a constant, decays exponentially [15] Thus, after the scrambling time, regardless of V and W , the correlator takes the following form This exponential decay is related to the rapid initial growth of the commutator, which becomes highly non-trivial at the scrambling time. For t t * the commutator is suppressed by the large-N term, and it is both small, and approximately constant. Notice that for chaotic quantum systems the behavior of the commutator at early times is similar to that of complexity; namely, both quantities are initially low (and approximately constant for t t * ), and later on undergo exponential growth, saturated at the scrambling time [10]. 6 The growth of the commutator is illustrative of the growth of complexity of W (t). As we later discuss, one way to see this would be to examine the relative state complexity.

Geometrical Interpretation of Complexity
Following Nielsen's approach [12][13][14], we now put forward a geometrical interpretation of complexity to illustrate its relation to gravity. Here, complexity is interpreted as "distance" on a unitary manifold. In this language, "distance" essentially means computational cost. Note that due to the assumed bulk/boundary Hilbert space equality, distances have to be preserved across the duality. With that in mind, the geometric approach will later be used to study the gravitational dual of relative state complexity of a small weakly-interacting auxiliary quantum system entangled to a large-N CFT. Usually, the Fubini-Study metric with its distance bound of suffices when talking about quantum states orthogonality. One issue, however, concerns the ease of saturating the bound. For the purpose of studying complexity, the Fubini-Study metric cannot adequately describe its exponential upper bound, and thus has to be substituted.
That is why for studying complexity we employ a non-standard 2 N -dimensional unitary manifold U (2 N ). Here, a time-evolving quantum state defines a trajectory s on the manifold, whose length naturally increases with time, corresponding to the state's increasing complexity. Similarly, for a pair of quantum states |ψ and |ψ , where |ψ , |ψ ∈ U (2 N ), each state defines its own trajectory on the manifold, where the distance between them linearly increases, and it is the geometrical analog of their increasing relative state complexity [20]. Moreover, the unitary manifold has intrinsic penalty factors which restrict quantum states from exploring more complex paths. Note that they are independent of the metric as a whole but rather depend on particular directions. Simple paths have O(1) penalties, while the more complex paths are exponentially suppressed by e N penalties. Obviously, the penalty factors are important because (i) they are related to the minimum possible complexity increase, associated with acting with a simple two-local gate (which geometrically can be interpreted as minimizing the geodesic length in the bulk), (ii) they define the local coordinates on the gravity side, and (iii) they define a natural notion of locality. Similar to the SYK model examined below, a generic object of interest in this framework is the evolution operator In a geometrical context, (2.11) formally reads where ← − P is a path-ordering operators and H(s) is a local Hamiltonian which parameterizes a path s on the unitary manifold where Y j (s) denotes the set of penalty factors which, when applied at every step (i.e. point) along the geodesic, control its path, and G i ≡ {g i } is a universal gate set. Proper choice of Y j ensures the computational task at hand is optimized. Geometrically, this translates to minimizing 7 the local geodesic on U (2 N ). Working in a unitary manifold, the complexity of U (t) is e N , which is to say that U (t) (here described as a point on U (2 N )) can explore an e N -dimensional state space. More precisely, we focus on the increase of complexity associated with applying U (t) to an arbitrary pure quantum state |ψ ∈ U (2 N ). Note that when we time-evolve a quantum state e −iHt |ψ (2.14) the corresponding complexity growth is independent of |ψ . Rather, the complexification is solely determined by the local Hamiltonian. Essentially, a time-evolving quantum state behaves like a non-relativistic particle moving across the unitary manifold. Where for a strongly-coupled large-N system, due to the chaotic dynamics, the distance traveled (i.e. increase of complexity) at the scrambling time (2.4) is sourced by the Lyapunov exponent (2.6), and hence grows exponentially where at late times t > t * it saturates to an evolution linear in N Evidently, the growth of the distance, traveled by a time-evolving quantum state on the unitary manifold, is (i) exponential for t ∼ t * , (ii) linear in N for t > t * , and (iii) a function of the local Hamiltonian Given the discrete time-step evolution of the random quantum circuit, when describing the trajectory s, spanned by the quantum state on U (2 N ), we need to specify how the local Hamiltonian acts at each time step ∆τ . For that purpose we define an instantaneous Hamiltonian H, i.e. a Hermitian operator which depicts the point-by-point evolution of the trajectory sH Here, an infinitesimal change along s corresponds to a simple unitary operation, i.e. acting with a two-local gate, and reads where geometrically, (2.19) can be expressed in the Schrodinger picture as, Fig. ( Despite being different than the classical Hamiltonian H, for the simplest case of an evolution operator U (t) that we consider (2.11), H = H. 8 Therefore, a time-evolving quantum state, expressed in terms of the instantaneous Hamiltonian, readsH where ρ(t) is the density matrix corresponding to the time-evolving state and j denotes the number of time steps ∆τ . Essentially, (2.21) provides a microscopic step-by-step (i.e. from j to j + 1) description of a time-evolving quantum state (2.14), Fig. 2 Following (2.13) and (2.21)-(2.23), the instantaneous Hamiltonian factorization yields To further illustrate the relation between complexity and geometry, namely the notion of distance on the unitary manifold, suppose we have a pair of arbitrarily close (i.e. low relative state complexity) quantum states |ψ and |ψ , where |ψ , |ψ ∈ U (2 N ), Fig. 1. Geometrically, the initial growth of complexity (dominated by the Lyapunov exponent) can be interpreted as the rapid divergence of the trajectories of |ψ and |ψ . Thus, the distance between the quantum states on U (2 N ) increases exponentially (2.15). As a result, the relative state complexity between the states is highly non-trivial at the scrambling time (2.1). Below we show that the large-N SYK model (which is chaotic and holographically dual to D=2 quantum gravity) and describes the evolution of a chaotic Hamiltonian yields an identical late-time growth, and geometrical interpretation of complexity.

Sachdev-Ye-Kitaev
Our goal is to further motivate the complexity and chaos evolution estimates and their geometrical interpretation. For this reason we proceed by very briefly sketching the geometrical approach to complexity in the standard large-N SYK model describing a chaotic Hamiltonian in the Lie algebra formalism. 9 Consequently, we demonstrate the results in Section (2.3) can easily be reproduced since the main objects have natural analogs within SYK. Similar to the holographic case, to have a well-defined geometrical approach to complexity in the SYK model, we need a basis which yields a notion of locality. We classify the Lie algebra of unitaries into two subgroups, i.e. "easy" and "hard" directions on the unitary manifold. A natural notion of locality is introduced by the simple (i.e. low complexity/computationally economical) generators {T i } in the Lie algebra, where {T i } = γ 1 γ 2 ...γ n , which are analogous to quantum gates in the quantum-information-theoretic approach. The Lie algebra generators are k-local, where for simplicity we set k = 2, and strictly penalize the (k > 2)-local ones. The k = 2 restriction simply means no generator can act on more than two gamma matrices at a time. Here, the gamma matrices {γ i } ∼ N which satisfy the Clifford algebra, where {γ α , γ β } = 2δ αβ , play the role of qubits. Similar to the discussion above (see Section (2.3)), locality implies the geodesic can only explore simple paths (i.e. sourced by k = 2 generators) on the unitary manifold, thus retaining its local minimum. Likewise, the restrictions on "easy" and "hard" directions are imposed by penalty factors which favor the use of k = 2 generators. That way, within SYK, we have a straightforward notion of locality on the unitary manifold which assures the geodesic is locally minimized. Therefore, considering (2.13), and assuming a universal two-local set of generators {T i }, the path of a geodesic on U (2 N ) is [9] For a detailed take on this approach, see [9] and the references therein.
where V i denotes the velocity terms. The unitary operator along the trajectory can thus be given as Here, assuming a given direction is not strictly penalized, the velocity terms dictate the path of the geodesic. This implies that we need to consider the velocities for every value of t, i.e. at each time step. Apparently, the velocities are equivalent to the instantaneous Hamiltonian in the holographic picture. In fact, we can make the relation even more precise and express the instantaneous Hamiltonian (2.18) in terms of the velocities in the tangent space as where only O(1) penalties are considered. This relation ensures the unitary manifold is well-defined at each point along the geodesic. Evidently, the unitary evolution of the geodesic on the unitary manifold is sourced by (i) the velocities which control the path, and (ii) the two-local Lie algebra generators which ensure locality. The relation between geometry and complexity in the SYK model can thus be schematically expressed as [8] C where G ij denotes the positive-definite bilinear metric form, and H is the Hamiltonian. Therefore, considering only the simple generators, the late-time growth of complexity reads [7,9] C(t) = n (E n t + 2πk n ) 2 1/2

(2.29)
Obviously, the behavior of complexity for a chaotic Hamiltonian in the large-N SYK model agrees with the holographic framework. Namely, at late times complexity increases linearly for time exponential in the number of degrees of freedom. Geometrically, a quantum state on the 2 N -dimensional state space moves with velocity given by the sum over E n , and its complexity is related to the distance traveled.

CA & CV dualities
In this Section we briefly review two conjectures relating holographic complexity to degrees of freedom in the gravitational dual.

Complexity=Action
In [21] Susskind et al. suggested the complexity of a CFT, living on the boundary of an asymptotically AdS spacetime, is dual to the action of a WDW patch in the bulk. The WDW patch is defined as the union of the past and future light cones of a spacelike hypersurface, anchored at some boundary time. One can also think of it as the region spanned by the union of all spacelike hypersurfaces (i.e. slices) anchored at some boundary time. In its most general form the CA-duality reads where the boundary complexity is dual to the action of the entire WDW region which extends deep within the AdS black hole interior. Usually, the bulk I W DW contains an Einstein-Hilbert action, and a York-Gibbons-Hawking boundary term. The CA-duality is equally well-defined for both, one-and two-sided AdS black holes. Suppose we have a CFT dual to a one-sided black hole in the bulk, and pick an arbitrary boundary time t. The state of the corresponding patch would be where H is a local Hamiltonian. Eq. (3.2) can be straightforwardly extended for the case of two entangled copies of a boundary CFT, dual to an eternal two-sided AdS black hole. In particular, picking t L and t R for the left and right AdS boundary, respectively, yields where the TFD state is given as here, |n L,R denotes the energy eigenstates, and β is the inverse temperature. Thus the conjectured CA-duality (3.1) suggests C (|ψ(t L , t R ) ) = I W DW π (3.5)

Complexity=Volume
Initially proposed in [23] the CV-duality 10 relates the complexity of a boundary CFT to the volume of a maximally-extended spacelike hypersurface behind the horizon where l AdS is the AdS radius. Behind the horizon the volume of the hypersurface has been shown to grow linearly like where S and T are the entropy and the temperature of the black hole, respectively.
As it was pointed out in [5,6], however, the CV-duality lacks the universality of the CA-duality since it requires hand-put length scale. In particular, the relation (3.6) is only valid assuming the black hole is large compared to l AdS . Otherwise, for black holes smaller than l AdS , (26) reads where r + is the horizon radius. Note that r + depends on the mass of the black hole, and thus has to be put ad hoc. Therefore, the CA-duality is considered more universal, and can easily reproduce the CV relation.

Holographic Relative State Complexity
In the current Section we propose a holographic interpretation of the relative state complexity of an auxiliary system of M weekly-interacting degrees of freedom coupled to a large-N CFT in a high-energy pure state, where M N . It has been argued that ER=EPR [4]. A pair of CFTs in a nearly maximally-entangled state, living on the conformal AdS boundary, are dual to an eternal two-sided AdS black hole with smooth geometry behind the horizon. That is, a pair of correlated AdS black holes connected by an ER bridge. As is well known, AdS black holes with radius ∼ l AdS do not evaporate due to the reflective conditions of the conformal boundary. However, as it was suggested by Raamsdonk in [3], weakly coupling a high-energy CFT (gravitationally dual to a one-sided AdS black hole) to an auxiliary system perturbs the boundary conditions, and the AdS black hole evaporates. This way, black hole degrees of freedom leak to the auxiliary system which now contains the radiation. If the auxiliary CFT is an exact copy of the original CFT, the two boundary theories are initially weakly correlated, and adiabatically evolve toward a TFD state (3.4). Here, we extend Raamsdonk's argument for the case of a large-N high-energy CFT, weakly entangled to a weakly-interacting auxiliary system of M degrees of freedom, and address the question: What is the gravitational dual to the relative state complexity of the auxiliary system? 11 On the AdS boundary, the combined system of a large-N high-energy CFT and a small auxiliary CFT of M degrees of freedom begins in a product state (4.1). By weakly correlating the two CFTs (4.3), the initial product state adiabatically evolves to a TFD state (3.4). In the bulk, this is dual to a one-sided AdS black hole which at early times is in thermal equilibrium with its environment. The black hole then begins to evaporate, and at the Page time becomes maximally entangled with its Hawking radiation. We will now examine the relative state complexity of the auxiliary system with respect to the identity 11 It should be noted that the dual AdS black hole is "young," meaning it has evaporated much less than half of its initial degrees of freedom. This is evident from the small dimensionality of the Hilbert space of the auxiliary system M N , and the initially low degree of entanglement to the CFT.
|φ(t 0 ) = I = 1, i.e. relative to its value at t = 0, where we will focus on the intermediate phase of the evolution before the TFD state is reached. Our claim is that this relative state complexity can be interpreted as being dual to an effective low-energy notion of computational cost (decoding task) in the bulk, namely to the minimal depth D min of the quantum circuit, required to decode the auxiliary CFT (i.e. most efficiently execute the operation (4.5) in the form (2.21)). The depth of a quantum circuit gives the complexity-per-qubit measure of the computational task at hand. In the random quantum circuit model we employ, using the depth yields the number of time steps j (or equivalently, the time) needed to carry out (4.5), and thus quantifies the relative state complexity. This measure is particularly useful when dealing with quantum systems composed of interacting qubits, especially in the complexity geometry approach. 12 More precisely, consider the following setup [3]. On the conformal boundary we begin with a large-N CFT in some high-energy pure state |ψ 0 , and introduce a small-M auxiliary system in its vacuum state |0 . The combined system is initially in a product state with a corresponding Hilbert space factorization of the form Suppose we now introduce an interaction Hamiltonian which weakly couples the two CFTs where O γ N and O γ M are locally defined operators which only act on their respective Hilbert spaces, and C γ N M denotes a family of coefficients. This would translate in the bulk to the one-sided AdS black hole starting to evaporate. Due to the transfer of modes, the auxiliary system is perturbed, and is no longer in the vacuum but instead in some typical state |φ . Moreover, the dimensionality of its Hilbert space H M monotonically increases, corresponding to the steady growth of the number of its degrees of freedom M due to the transfer of modes from the AdS black hole. Essentially, the adiabatically growing auxiliary system plays the role of a Hawking cloud for an evaporating black hole in asymptotically flat spacetime. For small t, the complexity of the auxiliary CFT is low C(|φ ) C max and due to the weak interactions of its degrees of freedom it grows linearly in M Geometrically, (4.4) implies the quantum state of the auxiliary CFT moves in a particlelike manner across the unitary manifold with the length of the geodesic linearly increasing 12 Using the minimal depth of a quantum circuit as a measure of the computational cost associated with executing a task, e.g. (4.5) resembles the Hartman-Maldacena tensor networks [25]. Tensor networks, like quantum circuits, have width and depth, and their evolution is very similar to that of circuits. Figure 2. An evolving quantum state |φ from the identity |φ(t 0 ) = I = 1 to some later state |φ(t 1 ) on a unitary manifold U (2 N ). The line depicts the minimal geodesic between the quantum state at the two instances, t 0 and t 1 . The distance between t 0 and t 1 corresponds to the relative state complexity, namely the computational cost associated with going from |φ(t 1 ) to |φ(t 0 ) (4.5).
The zoomed-in region depicts the discrete evolution dictated at each step by the instantaneous Hamiltonian (2.21) and bounded by the penalty factors (2.13).
(2.16). The perturbation of the auxiliary CFT due to (4.3) can schematically be given as |φ(t 1 ) = U |φ(t 0 ) . On the unitary manifold this simple operation corresponds to the geodesic travelling from |φ(t 0 ) to |φ(t 1 ) , Fig. 2, where the distance between the states gives the relative state complexity with respect to the identity. That is, it quantifies how much more computationally demanding |φ(t 1 ) is to decode. In other words, how many more steps j it would take. We suggest this increase of the relative state complexity of the auxiliary CFT is reminiscent of the Harlow-Hayden argument [19] regarding the increase of the computational cost associated with decoding the Hawking cloud. 13 The boundary relative state complexity between the identity and the perturbed state naturally increases with time. In the bulk, that is identical to saying the computational cost associated with decoding the small-M auxiliary CFT (i.e. execute the operation (4.5)) grows with time. Unlike in [19] where Alice's goal was decoding only subfactors of the Hawking radiation in order to verify entanglement with the black hole atmosphere, here the decoding task concerns applying unitary transformations to the auxiliary system to try to derive its initial state We therefore conjecture that the computational cost of implementing (4.5) can be interpreted as being gravitationally dual to [11] 13 Harlow-Hayden [19] argued that AMPS's conjectured violation of the equivalence principle after Page time is computationally unrealizable for astrophysical black holes since it requires complicated measurements on the emitted Hawking quanta the execution of which would take time exponential in the entropy.
whereH(s) is as in (2.21) given some appropriate (2.13), and D min denotes the minimum depth of the quantum circuit.
Since the auxiliary CFT plays the role of a Hawking cloud, we can apply similar analysis as in [19,24] to try to estimate (4.6). Generally, the computational task Alice faces scales like 2 M for m > 0, where m denotes the leaking degrees of freedom to the auxiliary system. She could, of course, take different approaches to try to decode the auxiliary CFT, and hence execute (4.5) efficiently. For instance, we imagine Alice could manipulate the degrees of freedom of the adiabatically growing auxiliary system, and engineer them into individual sets. She could then apply, in succession or in parallel, unitary transformation to the different sets in any arbitrary order she wishes. As it was demonstrated in [19], however, this procedure of limiting the unitary transformation to any particular group of degrees of freedom is especially complicated. Even more so, given m > 0, meaning the conformal AdS boundary conditions are changed, and hence degrees of freedom leak to the auxiliary system, multiple such limiting transformations have to be considered, further complicating the computation. Another decoding approach Alice could take is to make specific gates act on particular groups of degrees of freedom. Or similarly, connect different groups to specific gate subsets, and choose which groups to be acted on and when. Establishing any such connections would obviously require introducing extra degrees of freedom which scale as e M . Note that in [24] we proposed a novel measure of chaos in a strongly-coupled quantum system of many degrees of freedom (e.g. black hole) in terms of the complexity of the minimal quantum circuit necessary to decode the Hawking radiation, and we demonstrated that, due to the chaotic interior black hole dynamics, Alice cannot decode the Hawking subsystem (i.e. execute the operation (4.5)) faster than time exponential in the entropy. Furthermore, we showed that Alice can either act with a maximally-complex unitary operator or act with future precursor operators to the perturbed state, and count on extreme fine-tuning, where both options were argued to be computationally unreasonable for astrophysical black holes. The exponential growth at the scrambling time (2.4) of the minimum number of time steps j (as defined in (2.21)) required to implement (4.5), and thus calculate the relative state complexity, indicates the presence of chaos. We showed that by studying the circuit complexity, we can learn about the efficiency of the information processing, and the chaotic dynamics of the black hole interior. Furthermore, by using the circuit complexity as a measure of quantum chaos, we demonstrated that the Hawking radiation is pseudorandom. Namely, assuming there are information-carrying particles among the radiated thermal Hawking quanta, they cannot be decoded (i.e. be distinguished) in time less than exponential in the entropy, given Alice does not have infinite computational resources, and can only act on the emitted radiation. That is, the perturbed qubits are scrambled beyond recognition, given Alice has (i) subexponential available time to carry out the computation, and (ii) has limited computational power. We have thus made the case that the 2 k+m+r bound proposed by Harlow-Hayden [19] holds strong even for young black holes. 14 Putting the computational limitations aside (since they have been examined in detail in [19] and [24]), the main lesson here is that, in the bulk, decoding the small-M auxiliary system only gets harder with time. That is, the computational cost associated with |φ(t 1 ) increases, and that is simply because on the boundary, the relative state complexity between the identity |φ(t 0 ) and the perturbed state |φ(t 1 ) has a natural tendency to grow. Of course, one can look at the argument and ask: What if we set m = 0, and thus make M = constant? In other words, what if no modes leak into the auxiliary system? As it was demonstrated in the CA and CV conjectures, even if the number of degrees of freedom of the auxiliary system is held fixed, its complexity will still increase. Like entropy, complexity, statistically speaking, increases until C max is reached. So even for an AdS black hole which has saturated its coarse-grained entropy, dual to a boundary theory in thermal equilibrium after the scrambling time, complexity is much more likely to continue to increase for time exponential in the entropy due to its exponential upper bound.

Conclusions
In summary, we examined the case of a small auxiliary system of M degrees of freedom weakly coupled to a large-N high-temperature CFT, whose bulk dual is an evaporating onesided AdS black hole entangled to the Hawking cloud. We demonstrated that the natural linear increase of the relative state complexity of the auxiliary CFT, (4.4), with respect to the identity, i.e. between |φ(t 0 ) and |φ(t 1 ) could be interpreted as being dual to a lowenergy notion of computational cost (decoding task) in the bulk. That is, to the minimal depth of the quantum circuit required to decode the auxiliary CFT, namely to execute (4.5) in the form (2.21). In particular, the auxiliary system, playing the role of a Hawking cloud in the bulk, gets harder to decode with time, corresponding to the increasing relative state complexity of its boundary dual. Geometrically, we showed that this corresponds to a linear increase of the distance on the unitary manifold between the state's current and initial position, Fig. 2.