Remarks on Black Hole Complexity Puzzle

Recently a certain conceptual puzzle in the AdS/CFT correspondence, concerning the growth of quantum circuit complexity and the wormhole volume, has been identified by Bouland-Fefferman-Vazirani and Susskind. In this note, we propose a resolution of the puzzle and save the quantum Extended Church-Turing thesis by arguing that there is no computational shortcut in measuring the volume due to gravitational backreaction from bulk observers. A certain strengthening of the firewall puzzle from the computational complexity perspective, as well as its potential resolution, is also presented.


Introduction
Recently a certain conceptual puzzle in the AdS/CFT correspondence has been identified by Bouland-Fefferman-Vazirani (BFV) [1] and Susskind [2]. The puzzle is concerned with the growth of quantum circuit complexity in boundary CFTs and its dual on the bulk; the growth of the wormhole volume.
It is widely believed that finding the quantum circuit complexity of a quantum state is generically a difficult computational problem. Thus, there should be no efficient way of computing the complexity of the boundary CFT wavefunction. But the wormhole volume is a macroscopic quantity which appears to be easily measurable by bulk observers. This would suggest that a certain physical phenomena in quantum gravity may not be efficiently simulated on a quantum computer, violating the quantum Extended Church-Turing (qECT) thesis [3].
In this note, we propose a resolution of this puzzle. Any attempt to see the black hole interior by an observer always introduces significant gravitational backreaction to the underlying geometry. This invalidates protocols that would appear to measure the volume/complexity efficiently on the bulk, and thus the qECT thesis remains valid. The conclusion follows from the fact that a black hole scrambles quantum information unitarily and hence the infalling observer's backreaction disentangles the outgoing Hawking mode from the other side of the black hole completely. We also critically comment on the black hole complementarity approach to the complexity puzzle advocated by Susskind [2].
This note is organized as follows. In Section 2 and 3, we provide a brief review of the black hole complexity puzzle. In Section 4, we discuss the backreaction from the infalling observer. In Section 5, we present the resolution of the puzzle. We also present a refined interpretation of a certain thought experiment due to Hayden and Preskill. In Section 6, we present brief discussions. In Section 6.1, we present a certain strengthening of the firewall puzzle from the perspective of the qECT thesis.

Complexity
The quantum Extended Church-Turing (qECT) thesis is the following belief/physical principle [3]: -All the physical processes that obey fundamental laws of physics, including quantum gravity, are efficiently simulable on a quantum computer.
A thesis is a proposal for the sake of argument. It is not formulated as a precise mathematical conjecture and its correctness remains to be verified. If the qECT thesis is wrong, one could potentially speed up quantum computation by using quantum gravity effects 1 . The AdS/CFT correspondence is a prime setup to test the qECT thesis as it posits that all the physical phenomena in the bulk quantum gravity are encoded in boundary quantum mechanical systems and may be efficiently simulated on a quantum computer.
To derive the puzzle, we will need a contraposition of the qECT thesis within the AdS/CFT correspondence.
-Any physical processes that cannot be efficiently simulated in boundary CFTs cannot be efficiently simulated in bulk AdS quantum gravity.
To make the logic of this note clear, it is worth representing the statement of the qECT thesis schematically: bulk boundary Since the bulk consists both of quantum mechanics and gravity, it is reasonable to think that the bulk is stronger than (or equal to) the boundary in terms of the computational power 2 . The qECT thesis says that the AdS gravity does not provide any quantum computational speedup.
The central object of interest is the complexity of boundary wavefunctions in the AdS/CFT correspondence. The quantum circuit complexity C of a quantum state |ψ is the minimal number of few-body quantum gates that are required to create |ψ from some simple reference state such as |0 ⊗n .
While the precise definition of C depends on choices of elementary gate sets and other details, these subtleties will not be essential in demonstrating and resolving the complexity puzzle.
Recall that the two-sided eternal AdS black hole is conjectured to be dual to the thermofield double (TFD) state |TFD ∝ i e −βE i |ψ i L ⊗ |ψ i R . Here we will focus on black hole geometries arising from time-evolutions of |TFD under weak perturbations that can be treated as gravitational shockwaves. We will assume that thermodynamic properties of the black hole, such as the temperature, do not change by such perturbations. The following conjecture was proposed [4]: -The quantum circuit complexity C of boundary CFT wavefunction is dual to the wormhole volume V in the maximal volume slice.
In other words, C ≈ V in some appropriate normalization.
The C ≈ V conjecture has passed several non-trivial tests in a qualitative sense. To gain some insight, consider a time-evolved state |TFD(t) ≡ (e −iHt ⊗ I)|TFD . It is naturally expected that the complexity of |TFD(t) linearly increases in t when H is chaotic. While this statement is unproven, there is a large body of supporting evidence from various perspectives, such as quantum complexity theory [5] and pseudorandomness [6,7]. On the bulk, consider the time-evolved TFD state |TFD(t) = |TFD( t 2 , − t 2 ) and its dual wormhole in the maximal volume slice. Here we used the fact that The volume V indeed grows roughly linearly in time t for an exponentially long time, as in Fig. 1(a). Hence, after some proper normalization, we expect to have C ≈ V. Another justification of C ≈ V is obtained by invoking the tensor network representation of the spacetime [8].
In demonstrating the black hole complexity puzzle, we will be interested in the computational difficulty of determining C from |ψ , not C itself. Suppose that some unknown wavefunction |ψ is given. Without prior knowledge on how |ψ was prepared, determining its quantum circuit complexity C is a difficult computational problem. In the most generic setting of the problem, we could at best try to apply quantum gates to see if |ψ returns to some simple state. Since there are exp O(C) nearly-orthogonal states with complexity C, the complexity of estimating C is expected to be ∼ exp(C).
-The quantum computational complexity of estimating C is ∼ exp(C) in general, and thus is a difficult computational problem.
While unproven, there are quantum computational complexity theoretic arguments supporting this conjecture, see [1] for instance.
They provided an argument suggesting that it is indeed difficult to estimate C even if one knows H since it is difficult to distinguish the above ensemble from some random ensemble of much higher complexity.
Here it is important to assume that the choice of observer. Unless otherwise stated, we shall focus on |TFD(t) without insertion of multiple shockwaves for simplicity of discussion.
Combining the above conjecture with another one C ≈ V, we arrive at the following conjecture: -Estimating V from generic boundary CFT wavefunctions is a difficult computational problem.
Below is a side comment. There is a certain heuristic relation between two-point correlation functions and the complexity of estimating C. Two-point correlation functions between two sides of a black hole decay exponentially as the separation increases. This suggests that measurement of two-point functions by bulk observers, which would go as ≈ e −γC , can give an estimate of the wormhole length where γ is some positive constant. When the complexity C is high, the measurement outcome is too weak, and one would need ≈ e +γC samples to make a reliable measurement of two-point functions. This is consistent with the fact that the complexity of estimating C is conjectured to be ≈ e γ C for some constant γ .
Turing the argument around, the exponential growth of the complexity of estimating C would suggest the exponential (not polynomial) decay of two-point functions with γ γ . It is interesting that bulk field theories can put some restrictions on the complexity of estimating C 3 . If the bulk theory is somehow gapless with polynomially decaying correlation functions, the complexity of estimating C would be at most poly(C).

Puzzle
In the light of the qECT thesis, estimating the wormhole volume V should also be computationally difficult since estimating C is difficult. This is, however, strange. The wormhole volume is roughly proportional to the length of the wormhole which is a simple macroscopic quantity that appears to be easily computed or measured. This would suggest that estimating V may actually be a computationally tractable problem for bulk observers, violating the qECT thesis.
BFV sharpened this observation by proposing a certain protocol to measure the wormhole volume by using multiple observers who live on multiple copies of the system. Imagine that one somehow populates the black hole interior with many observers along the wormhole, and see if pairs of observers can send signals to one another before reaching the singularity ( Fig. 1(a)). The number of successful signal transmissions will be a coarse estimate of the wormhole volume.
One potential problem in the BFV protocol is the use of multiple observers who are causally disconnected. Another issue is that, to place observers in the interior, one would need to know the shape of the wormhole a priori, which implicitly assumes prior knowledge of V. At the same time, however, the description of the protocol in [1] is rather brief, and we are unsure if we have addressed its full intent 4 .
Susskind presented another version of the puzzle which is free from these problems. Suppose that some boundary wavefunction |ψ(t) undergoes time-evolution. Since it is difficult to estimate the complexity of |ψ(t) , it should be also difficult to judge if C(t) is increasing or decreasing at given time.
Combining this observation with the C ≈ V conjecture, the following conjecture is obtained.
-Judging if V(t) is increasing or decreasing from boundary CFT is computationally difficult.
To demonstrate the puzzle based on this observation, let us look at two classes of boundary wavefunctions. First, consider the time-evolution of the TFD state |Ψ(t) ≡ |TFD(0, −t) at t L = 0 and t R = −t. As we have seen, C(t) of |Ψ(t) increases since V(t) increases in time ( Fig. 1(a)).
Next, consider another wavefunction at t L = 0 and t R = −t where, at some negative time t L = −τ , a small perturbation is applied ( Fig. 1(b)): It has been pointed out that C(t) of |Φ(t) actually decreases for τ t. On the bulk, this can be verified by directly computing the volume V, see [4] for details. On the boundary, this decrease can be easily seen by drawing a quantum circuit diagram as in Fig. 1(c). Here we have prepared the initial state |Φ(0) by inserting a perturbation on the thermofield double state on the right hand side cancels the e iH R τ , and hence the complexity C(t) actually decreases until t = τ .
The take-away from the above analysis is that complexity of wavefunctions can decrease or increase depending on whether a perturbation is applied on the left side or not. This is, however, strange from the bulk observer's perspective. An observer simply needs to jump into the black hole and see if she will be hit by a gravitational shockwave which lets her allow to judge the sign of dC dt . Once again, the qECT thesis seems to be violated.
Several flaws in the above arguments can be immediately identified. Introducing a bulk observer behind the horizon corresponds to high complexity quantum operation on the boundary. Namely, when approaching near the horizon, an infalling observer herself induces significant gravitational backreaction to the underlying geometry. These effects often invalidate bulk effective descriptions, and hence should be examined carefully, which is exactly what we will do in the next section. Also it is unclear why an infalling observer from the right hand side is able to see a shockwave from the left hand side while two sides are not coupled. Later we shall indeed see that an observer will not be able to see the shockwave due to the backreaction caused by the observer herself.
In the reminder of the note, we argue that measuring V or dV dt on the bulk is not necessarily a computationally easy task by examining the effect of backreaction by infalling observers.
For convenience of readers, we summarize the argument of the complexity puzzle 5 . 1). On the boundary, quantum circuit complexity C (or dC dt ) is believed to be not efficiently computable.
2). On the bulk, the wormhole volume V (or dV dt ) seems to be efficiently measurable by infalling observers.
3). The wormhole volume V and the quantum circuit complexity C are roughly proportional to each other. 4). The qECT says that, as C (or dC dt ) is not efficiently computable, V (or dV dt ) should not be efficiently measurable either.
We will argue that 2) is wrong. Here 4) is the statement of the qECT thesis while 1) and 3) are widely-accepted conjectures.

Backreaction
The main goal of this section is to discuss the backreaction from an infalling observer on the black hole interior geometries. Our strategy is to find interior partner modes that are entangled with outgoing modes in boundary CFTs. By studying how entanglement structure changes by the addition of the infalling observer, we can deduce bulk geometries with appropriate gravitational backreaction. This section is a short summary of [11,12].
For simplicity of discussion, we represent the black hole as a quantum system of n qubits. As the initial state of the black hole, consider a generic maximally entangled state between the left side B and the the right side B: where K is an arbitrary unitary operator. Imagine that some measurement probe (or an infalling observer) A is dropped into the black hole at time t = 0 as in Fig. 2(a). Rather than keeping track of the outcomes for all the possible input states on A, it is convenient to append a reference system A which is entangled with the probe A and forming EPR pairs After the time-evolution by some unitary dynamics U , the system evolves to where D is the outgoing mode and C is the remaining black hole. Horizontal lines represent EPR pairs.
Here black dots represent factor of 1/ √ d R in a subsystem R for proper normalization of |EPR RR . A dotted line divides the left and right sides of the black hole.
In the absence of the infalling observer A, the outgoing mode D is entangled with some mode R D For more details on the definition of scrambling via OTOCs, see [13].
The inclusion of the infalling observer has drastic effect on entanglement structure due to the scrambling dynamics of the black hole. The following statement was proven in [14]. Suppose that the system is scrambling in the above sense and d A d D 6 . Then, the subsystems D and B in |Ψ are decoupled (not entangled): where the error is O . Hence, we arrive at the following conclusion: -The outgoing mode D is no longer entangled with the right hand side B. Instead D is entangled with CA. Thus the entangled partner mode is found exclusively on the left hand side.
An immediate, yet important implication of this conclusion is that the construction of the interior partner mode is independent of the initial state of the black hole (or the unitary K in Eq. (3)). In fact, the same expression of the interior partner mode works for any initial state of the black hole, be it two-sided or one-sided since it is supported exclusively on the left hand side without using degrees of freedom from B. Hence, infalling observer's backreaction enables us to obtain state-independent construction of interior partner operators 7 .
The bulk interpretation of the above disentangling phenomena can be obtained by treating the backreaction from an infalling observer as a gravitational shockwave ( Fig. 2(b)). In the absence of the infalling observer, given the outgoing mode D, interior partner operators can be constructed by time-evolving a corresponding mode on the right hand side. This mode, constructed exclusively on the degrees of freedom on the right hand side B, is denoted by R D . We now include the effect of the infalling observer A as a gravitational shockwave and draw the backreacted geometry where the horizon is shifted as depicted in Fig. 2(b). If the time separation between the outgoing mode D and the infalling observer A is longer than or equal to the scrambling time, the interior mode D, which is distinct from R D , can be found across the horizon.
One successful application of the disentangling phenomena by an infalling observer is the resolution of the firewall puzzle. Recall that the outgoing mode D was initially entangled with some degrees of freedom R D in B. Assuming the smooth horizon, an infalling observer Alice would see an interior mode D which is entangled with the outgoing mode D. This, however, leads to a contradiction because D is 6 See [11] for discussions on cases where the time separation is shorter than the scrambling time as well as physical interpretation of the condition dA dD. 7 It is worth emphasizing, however, that the construction depends on the initial state of the infalling observer, and thus is observer-dependent. See [12] for details. also entangled with R D . If we would think that D remains entangled with R D , then D and D would not be entangled, leading to possible high energy density at the horizon.
Then the resolution of the firewall puzzle is immediate. When an infalling observer jumps into a black hole, the outgoing mode D is disentangled from R D due to her own backreaction. She will be able to observe the interior mode D which is distinct from the original partner mode R D . We will revisit this resolution by strengthening the firewall puzzle from the perspective of the qECT thesis in Section 6.

Resolution
We begin by arguing that Susskind's protocol for measuring dC dt cannot be performed. In his protocol, an infalling observer from one side jumps into the black hole and see if she will be hit by a shockwave from the other side or not. As is already mentioned, it is strange to expect that the infalling observer could see the shockwave when two sides of the black hole are not coupled 8 . The key to resolve this misunderstanding is to consider the effect of backreaction by an infalling observer as in Fig. 2(b).
Assume that the shockwave was created by exciting the boundary mode R D . Since this mode R D was initially entangled with D, the infalling observer would expect to be hit by the entangled partner mode of D behind the horizon. Due to her own backreaction, however, D will no longer be entangled with R D once the infalling observer jumps into the black hole. Inside the black hole, she will just see the interior modeD which is distinct from R D . Hence the infalling observer will not be able to see the shockwave from the other side .
Here we assumed that two sides of the black hole are not coupled. In order for two signals/observers from opposite sides to meet inside the black hole, two boundaries need to be coupled in an appropriate manner. By using the traversable wormhole phenomena [16] or the Hayden-Preskill decoding protocol [13], two observers may be able to see each other and then travel to the other sides of the black hole.
To implement these protocols, however, one needs to reduce the boundary CFT wavefunctions back to low complexity states (such as the TFD state), so there is no shortcut in these approaches. It is unclear to us if there would be a simpler way to make two observers meet inside the black hole. To influence two decoupled degrees of freedom deep in the bulk, it is natural to expect that high complexity quantum operations are needed on boundary CFTs.
Next let us turn to BFV's protocol. Here we shall focus on a certain subroutine of their protocol.
BFV's protocol requires a pair of observers who send signals to each other inside the black hole. Since two observers from opposite sides cannot communicate as discussed above, they should come from the same side of the black hole. Let us imagine that one observer (Alice) jumps into the black hole, and then the other observer (Bob) jumps later. Upon crossing the horizon, Alice sends a signal to Bob.
Can they communicate? Hayden and Preskill asked exactly the same question in [17]. Their argument goes as follows. If the time separation ∆t between Alice and Bob is longer than the scrambling time, Alice needs large amount of energy, possibly larger than the Planck energy, to send a signal to Bob.
Otherwise, Bob will reach singularity before Alice's message arrives. Hence two observers will not be able to communicate with each other unless they introduce drastic backreaction to the underlying geometry. This suggests that there seems to be a fundamental upper bound on efficiently measurable volume or length near the horizon.
Here we would like to present a modern perspective on the Hayden-Preskill's observation by considering the effect of backreaction by Alice. Suppose that Alice was planning to send a signal by using the mode S behind the horizon, which is the partner mode of S (Fig. 3(b)). Once Alice crosses the horizon, a new partner modeS is dynamically created. Hence Alice cannot send a signal to Bob by using the original partner mode S. In order to avoid backreaction, Alice would need to use the interior mode S which is not close to the horizon. However this means that the time separation ∆t between Alice and Bob is significantly shorter than the scrambling time.
While we have arrived at the same conclusion as Hayden and Preskill's, there is a subtle (but important!) refinement in our argument. The reason why Alice cannot communicate with Bob is because Alice has intersected with the mode S, and as a result, S is disentangled from S. Namely, the disentangling phenomena occurs automatically, regardless of whether Alice attempts to send a signal to Bob with trans-Planckian energy or not.
Finally, we critically comment on Susskind's proposal for a resolution of the complexity puzzle [2].
Susskind proposed that the qECT thesis may be violated behind the black hole horizon, but the violation should not be communicated to the outside efficiently. This proposal is in accord with the black hole complementarity; one chooses not to be bothered by the violation of the qECT thesis if the violation cannot be ever found out. As we have already demonstrated in the main body of this note, backreaction from infalling observers invalidate any shortcuts to measuring the volume of the wormhole. Hence we do not need to invoke the black hole complementarity to save the qECT thesis and resolve the puzzle.
The effect of gravitational backreaction prevents the violation from happening. It is wrong to say that the violation is invisible; the violation simply did not happen.
Another counterargument can be obtained by actually saving the infalling observer from a black hole. Recall that an infalling observer in a two-sided black hole can be saved from the black hole by quantum operations which act solely on one side of the black hole in principle [11]. The quantum circuit complexity of quantum operations is independent of the complexity of the initial state (i.e. the wormhole volume V of the two-sided black hole). Hence the infalling observer can report her experience after performing quantum operations with a complexity much smaller than V. This does not invalidate the qECT thesis; the rescued observer will not give us any useful information on the wormhole volume V or dV dt as she was not hit by the shockwave.

Discussions
We have presented a possible resolution of the black hole complexity puzzle by considering the effect of gravitational backreaction by an infalling observer. Our observation suggests that there is no shortcut in measuring the volume of the wormhole, and thus the qECT thesis remains valid . We also revisited the thought experiment by Hayden and Preskill who asked whether two infalling observers from the same side of a black hole can communicate with each other or not. Namely we presented a refinement of their argument (but with the same conclusion) by examining gravitational backreaction without invoking the black hole complementarity.
Our results also provide a resolution on the unboundedness puzzle concerning the diverging Hilbert space dimension of the black hole interior and the apparent violation of the area law in the maximal volume slice. Namely, our argument suggests that volume-law degrees of freedom can be regulated to give an area-law entropy in a dynamical manner. We speculate that a similar reasoning may resolve the puzzle concerning Wheeler's "bag of gold" spacetime. It is an interesting future problem to make this perspective more quantitative. We also believe that proper understanding of such phenomena will be useful in better understanding the physics of de Sitter space. Relatedly, it will be interesting to ask if the qECT remains valid for quantum gravity in de Sitter space.
Recently Herman Verlinde proposed an interesting argument on a possible loophole in the ER = EPR slogan [18]. The idea is that bulk effective theories emerge after projecting the system into a certain codeword subspace, and the preexisting quantum entanglement between two sides, as well as the complexity, is not essential for the wormhole geometry. We believe that our treatment of backreaction by infalling observers provides explicit construction of the projection operator in a dynamical manner.
It is an interesting future problem to understand dynamical emergence of bulk effective theories.
BFV demonstrated the complexity puzzle in a slightly different manner by considering the complexity of the bulk-boundary dictionary and arrived at the following conclusion; 1) the qECT thesis is violate or 2) the complexity of the holographic dictionary is high. In our argument, this no-go result has been avoided by the fact that the dictionary is observer-dependent and thus is not invariant. Indeed, the expression of the interior partner mode depends on the initial state of the infalling observer.

Extended firewall puzzle
Motivated by the black hole complexity puzzle, we would like to point out another interesting challenge to the qECT thesis in the context of the firewall puzzle. The challenge concerns a certain puzzle on the decoding complexity of interior partner modes. We view this as an extension of the firewall puzzle from the computational complexity perspective.
The gravitational backreaction by an infalling observer provides a potential resolution of the firewall puzzle since the outgoing mode is decoupled from the early radiation due to scrambling dynamics of a black hole. As such, the fact that the infalling observer can cross a smooth horizon and see interior partner modes behind it does not lead to any inconsistency from the perspective of the monogamy of quantum entanglement [11]. From the computational complexity perspective, however, we find some trouble in this scenario. From the boundary viewpoint, finding the expression of interior partner operators appears to be computationally difficult especially when the Hilbert space size accounting for interior modes is large and OTOCs have decayed to asymptotic values at late times. The best known algorithm for interior reconstruction at late times uses the Grover search algorithm whose runtime is proportional to the Hilbert space size of the interior modes (hence is exponential in the number of qubits) [13] 9 . Yet, the infalling observer can see the interior Hawking modes by simply jumping into a black hole and crossing the smooth horizon without doing any difficult quantum computation! This leads to an apparent violation of the qECT thesis.
Below we propose a possible resolution of the extended firewall puzzle. When the time separation ∆t between the infalling observer and the outgoing mode is longer than the scrambling time t scr , the infalling observer will encounter the interior partner degrees of freedom near the horizon, possibly at the Planck length distance (or less) from the horizon. Such interior modes at short distance scale should not be visible to infalling observer since these would be realized as a part of the geometry itself, instead of matter fields propagating freely on the geometry, due to some short distance quantum gravity effect.
Hence it is reasonable to assume that interior partners at short distance scales are computationally difficult to decode for the infalling observer too. On the other hand, when the time separation ∆t is shorter than t scr , one may utilize decoding protocols based on traversable wormhole effects to efficiently construct interior partner operators as in [16]. These protocols utilize the fact that black holes are the fastest scrambler with the Lyapunov exponent 2π β and work reliably only when a black hole scramble quantum information coherently until ∆t ≤ t scr . In the bulk, this corresponds to the fact that the infalling observer will encounter these interior partner modes away from the horizon, and can observe them easily. Hence, the qECT thesis will remain valid.
A resolution of the original firewall puzzle, in a sense of the monogamy of quantum entanglement, required the fact that a black hole scrambles quantum information. This resolution, however, is unsatisfactory as it did not utilize the fact that a black hole scrambles quantum information in the fastest possible manner. After all, a piece of burning wood will scramble quantum information at late times too! The extended firewall puzzle, strengthened by the qECT thesis and the computational complexity, is intrinsic to black holes and naturally calls for the very fact that a black hole is the fastest scrambler.
Namely, the separation of the computational complexity in reconstructing interior operators before and after the scrambling time results from the fast scrambling nature of the black hole. On the bulk, these correspond to different physical objects; the matter and the geometry.