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Complexity and Boost Symmetry

Ying Zhao

TL;DR

The paper shows that holographic complexity time dependence is governed by the Rindler-like boost symmetry across horizons, with linear growth arising from this symmetry and controlled symmetry breaking by shocks yielding abnormal dynamics. It develops a two-tape interior picture where future and past tapes store forward and backward segments of the minimal circuit, and identifies tape working versus locked states with horizon smoothness and firewall formation. By analyzing shockwave geometries and time-dilation factors across horizons, the work connects complexity evolution to firewall physics and provides a diagnostic for interior smoothness based on how complexity changes with boundary time. The results tie together tensor-network pictures, bulk causal structure, and boundary notions of complexity, offering a unified view within classical GR and standard holographic complexity proposals, while suggesting boundary interpretations and future directions on sub-AdS locality.

Abstract

We find that the time dependence of holographic complexity is controlled by the Rindler boost symmetry across the horizon. By studying the collision energy experienced by an infalling object, we see the breaking of this boost symmetry is closely related to firewalls, which in turn shows the connection between the time dependence of complexity and firewalls. We further identify the black and white hole interiors as two tapes storing different parts of the minimal circuit preparing the state. Depending on whether the quantum gates are being laid on the tape at a particular moment, each tape can be in two states: working, or locked. We interpret the existence of firewalls as the locking of tapes.

Complexity and Boost Symmetry

TL;DR

The paper shows that holographic complexity time dependence is governed by the Rindler-like boost symmetry across horizons, with linear growth arising from this symmetry and controlled symmetry breaking by shocks yielding abnormal dynamics. It develops a two-tape interior picture where future and past tapes store forward and backward segments of the minimal circuit, and identifies tape working versus locked states with horizon smoothness and firewall formation. By analyzing shockwave geometries and time-dilation factors across horizons, the work connects complexity evolution to firewall physics and provides a diagnostic for interior smoothness based on how complexity changes with boundary time. The results tie together tensor-network pictures, bulk causal structure, and boundary notions of complexity, offering a unified view within classical GR and standard holographic complexity proposals, while suggesting boundary interpretations and future directions on sub-AdS locality.

Abstract

We find that the time dependence of holographic complexity is controlled by the Rindler boost symmetry across the horizon. By studying the collision energy experienced by an infalling object, we see the breaking of this boost symmetry is closely related to firewalls, which in turn shows the connection between the time dependence of complexity and firewalls. We further identify the black and white hole interiors as two tapes storing different parts of the minimal circuit preparing the state. Depending on whether the quantum gates are being laid on the tape at a particular moment, each tape can be in two states: working, or locked. We interpret the existence of firewalls as the locking of tapes.

Paper Structure

This paper contains 17 sections, 40 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Time dependence of complexity and boost symmetry
  3. Known results about holographic complexity
  4. Linear increase of complexity: Boost symmetry across the horizon
  5. Breaking boost symmetry: Abnormal time dependence of complexity
  6. Time dependence of complexity as a diagnosis of good interior
  7. Quantum circuit and the smoothness of horizons
  8. Future and past tapes
  9. Two tape picture: Why are there both future and past horizons?
  10. Tape working and the smoothness of horizons
  11. Conclusion
  12. Complexity of a precursor
  13. Switchback from multiple precursors
  14. Breaking spatial translation symmetry: Localized shocks
  15. Collision energy and time dilation factors
  16. ...and 2 more sections

Figures (27)

  • Figure 1: Boost symmetry across horizon. Schwarzschild time inside the horizon synchronizes with Schwarzschild time outside.
  • Figure 2: Product of multiple precursors
  • Figure 3: Bulk duals of $|\psi(t_R,t_L)\rangle$ and $|\psi(t_R+\Delta t, t_L)\rangle$.
  • Figure 4: Increase of WDW patch as right time increases from $t_R$ to $t_R+\Delta t$.
  • Figure 5: Two sided black hole with one spherically symmetric shockwave
  • ...and 22 more figures