SciPost Submission Page
From black hole interior to quantum complexity through operator rank
by Alexey Milekhin
Submission summary
| Authors (as registered SciPost users): | Alexey Milekhin |
| Submission information | |
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| Preprint Link: | scipost_202601_00039v1 (pdf) |
| Date submitted: | Jan. 16, 2026, 7:45 p.m. |
| Submitted by: | Alexey Milekhin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
It has been conjectured that the size of the black hole interior captures the quantum gate complexity of the underlying boundary evolution. In this short note we aim to provide a further microscopic evidence for this by directly relating the area of a certain codimension-two surface traversing the interior to the depth of the quantum circuit. Our arguments are based on establishing such relation rigorously at early times using the notion of operator Schmidt rank and then extrapolating it to later times by mapping bulk surfaces to cuts in the circuit representation.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
I would like to thank the referees for very stimulating questions. There is a considerable overlap in their questions, so let me list all changes in one section.
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I tried to make the Introduction more clear by stating what assumptions I invoke about the underlying physical system and make the structure of the argument more clear. Inequalities (2) [between entanglement entropy and complexity] are established rigorously for any discrete circuit evolution and are valid beyond holography. Then I invoke the standard holographic dictionary for obtain (3) [area of RT surface and entanglement entropy]. Together, (2) and (3) at early times lead to (1), which is the main statement of the paper. This also allows one to by-pass involved discussions of UV-regularization of field-theoretic complexity mentioned by one of the reviewers. Finally, I give heuristic arguments why (1) must hold for later times.
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I made some sloppy statements more sharp: I provided the exact value of $r$ for brickwork circuits. Also the approximate sign in (6) (previously (5)) was changed to equality sign. [originally, it meant to symbolize that there is gauge freedom in the decomposition as well as possible omission of very small singular values, but it is too vague and not really needed for the argument]
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Regarding the tightness of the bound: it is true that in a general situation von Neumann entropy and log of the rank might differ considerably. Surprisingly, holographic complexity and the area of HM surface has the same asymptotic growth: $t/\beta$ up to a constant. So the bound is very close to saturation. I offer more comments on this in the Discussion section.
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Finally, prompted by one of the referees I discuss the situation with Python's lunch. My conclusion is that the arguments in this paper are sensitive to the presence of the bulge but do not produce exponentially large (in $1/G_N$) factors in the conjecture. It is easy to understand why: the present argument only cares about the number of gates, all of the are assumed to have equal weight. Whereas the Python's lunch conjecture argues that non-unitary gates are very expensive due to post-selection. So, strictly speaking, the lower bound in this paper remains valid, but becomes very loose.
List of changes
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2026-1-25 (Contributed Report)
The referee discloses that the following generative AI tools have been used in the preparation of this report:
Used to correct grammar of report
Weaknesses
1- The author elaborates the consequence of having a non-unitary time evolution operator and its embedding into unitary ancilla systems. This significantly weakens the claim of having the HM surface to be described by quantum complexity.
2- The author claims that the non-unitary factor $e^{-\beta H/2}$ removes the UV degrees of freedom in a hand-wavy manner. The issue of UV divergence has to be addressed in a concrete way to make progress toward establishing an identity between quantum complexity and black hole interior geometry. In particular, the presence of the factor $e^{-\beta H/2}$ dismisses the discussion about quantum complexity.
3- Finally, the bound established by the author is rigorous at early times but extrapolated to late times by arguing that the late-time HM surface continues to exhibit linear time evolution. This is not well-defined as $\ln \chi$ saturates and the reader will be confused in what sense the bound is still valid based on the inequality in Eq(2) (and Eq(1)).
Report
Due to the above-mentioned reasons, this paper is seen to be remained as a proposal rather than a concrete justification on quantum complexity=black hole interior geometry as there are several major challenges that remain unanswered in this manuscript. In short, the argument of the manuscript is going in circles. On one side, there is these UV degrees of freedom that needs to be suppressed by the non-unitary factor $e^{-\beta H/2}$ and on the other side, a direct connection between unitary quantum complexity and black hole interior geometry needs to be established. I recommend that the author consider submission to SciPost Physics Core.
Recommendation
Accept in alternative Journal (see Report)
Strengths
Weaknesses
Report
As far as I can see, my point 3 about UV divergence, is only addressed in one place, on page 2, where the authors claims that to sidestep the issues related to UV divergences, he will use the entanglement entropy instead. But the entanglement entropy is also UV divergence in QFT (even if we are much more familiar with how to regulate it). For the rest of the paper, the authors discusses discrete circuits and QFTs (everytime he is doing holography, the quantum system is a CFT) interchangeably, and this makes it impossible to follow the logic.
Recommendation
Reject