Random Circuits in the Black Hole Interior

This is a very interesting paper. The authors gave a novel construction of long wormholes supported by matter and rigorously established the complexity property of an ensemble of such states. I have one minor comment. It would be helpful to compare the construction given in this paper and that of figure 7 in https://arxiv.org/pdf/1312.3296.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

- Papers explores the black hole interior, which is an important open question in holography.

- Very well written.

Some aspects related to the setup are confusing.

This is an interesting paper that attempts to quantify the extent to which the black hole interior describes randomness in the CFT state. Previous considerations studied the time-evolution of the thermofield-double state, and related the growth of the wormhole length in the black hole interior to the complexity of the quantum state, which increase under time evolution. However, these ideas are hard to make precise due to the challenge of making precise definitions of complexity in CFTs. Here, the authors explore another idea. They consider a family of TFD-like states, which are TFD states perturbed by sources for simple (low-$\Delta$) operators. The operators are inserted on particular time-contours to produce states of constant energy, even with the insertion of operators, and the sources are taken to be random with a Gaussian time-organized distribution. The averages gives rise to Einstein-Rosen caterpillars, modifications of the TFD state where the wormhole length is larger, and the inside geometry contains matter insertions correlated on some length-scale.

The authors show this ensemble of states, once the Gaussian average is performed, gives a quantum state $k$-design for true Haar random states of the CFT. This means that at large effective time of the circuit, the ensemble of states becomes k-copy indinstinguishable from a Haar random state.

This is a great paper and a useful addition to the literature. I would be happy to recommend it for publication, but there are a set of confusing points (most of them related to their setup) that the authors should first clarify. I list them below;

1. Above (1.5), the authors talk about the "difference" between $\rho_{eq}$ and the thermal state. Is this meant to be a precise statement or simply a statement that one difference between the two states is the variance in energy. Moreover, the energy variance is an important piece of the thermal states, and is captured by the specific heat. Should one not be worried that the ensemble of states have a different specific heat? I could not find a discussion of this in section 2. The authors should clarify.

2. I am confused about FIg. 5 and the way to cut it to prepare the state that the authors want to study. The idea from Fig. 3 is very clear, one starts adding sources at $t=0$ and this modifies the state and makes matter start to fall in. In Fig. 5, where should one cut the Euclidean contour to produce the state that we want to study? Following the idea of Fig. 3, I would have said right after the last time fold on contour 1. But the words around here, and also equation (2.8), make it look like they would cut Fig. 5 in the two places where $\beta/2$ is written. This is confusing, and the authors should clarify this point.

3. I also do not understand why there is no matter outside the horizon for the states described by the authors. Do they claim this is true for a single realization, or is this a property of the ensemble? If it is true for a single realization, I do not understand why. Inserting sources for matter fields always comes with tails. Even if you insert them in the middle of the Euclidean contour that prepares the TFD state. These tails could be suppressed by considering very heavy operators $O_\alpha$ with large scaling dimension, but I did not think this is what the authors were doing. If it is a property of the ensemble, then I would like to understand what can be said about the variance of the matter fields outside the horizon, even if they vanish on average. The authors should clarify these points.

4. The caption of Fig. 6 is also confusing. The authors write the operator $e^{-\tau H_{eff}} \ket{TFD}$. I think here they should not write the ket state, if they talk about a property of an operator. There also seems to be some confusion in notation between $t$ and $\tau$ here). Is $\tau$ not the same as $t$ in (2.15)? Finally, I do not understand why the authors call this geometry a Euclidean wormhole. Is there not a single Euclidean boundary for the geometry $M$? This definitely seems to be the case for the top left figure. The authors should clarify these points.

5. I am also confused about the statement that the authors want $K\sim N^2$. I thought $K$ labelled the number of distinct operators that one would insert. In bottom-up models, one considers a finite number of matter fields. Even in top-down models, there can formally be an infinite number of matter fields due to the KK reduction on internal manifolds, but one should be very careful in trying to scale the number of fields with $N$ and still trusting supergravity. This is an important distinction with the SYK model that has $N$ fermions to play with, which is often ignored in its application to holography due to the dominance of the Schwarzian dynamics, but here it is important. If the authors really need $K\sim N^2$, I am confused whether one can ever apply the analysis of their paper to any realistic example of AdS/CFT in $d>1$.

6. Above (2.21) the authors mention that the isometry is fake and only emergent after taking the ensemble. Can the authors comment on the variance of this isometry? If the variance is small, I suppose then that all realization themselves have an approximate isometry. If it is large, then the isometry should not be trusted.

7. The scaling with $k$ found by the authors comes from the number of excited states, which is factorial in $k$. The authors seem to not make a big deal of this, and treat a factorial growth essentially as an exponential growth, but I would argue that it is much faster. Of course, if you take a log, then the enhancement is only logarithmic but I think the authors should discuss this enhancement at least a little bit more than what they do, which is currently only in the form "up to log corrections" or "approximately linear in k". Does is not have any physical importance?

Ask for minor revision