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  • Letter
  • Open Access

Complexity and the Hilbert space dimension of 3D gravity

Vijay Balasubramanian1,2,3, Rathindra Nath Das4,5,6,*, Johanna Erdmenger4, Jonathan Karl4,†, and Herman Verlinde7

  • *Contact author: das.rathindranath@uni-wuerzburg.de
  • †Contact author: jonathan.karl@uni-wuerzburg.de
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Phys. Rev. D 114, L081901 – Published 5 October, 2026

DOI: https://doi.org/10.1103/wklh-xqbr

Abstract

A central problem in formulating a theory of quantum gravity is to determine the size and structure of the Hilbert space of black holes. Here we use a quantum dynamical Krylov complexity approach to calculate the Hilbert space dimension of a black hole in (2+1)-dimensional anti–de Sitter space. We achieve this by obtaining the spread of an initial thermofield double state over the Krylov basis. The associated Lanczos coefficients match those for chaotic motion on the SL(2,R) group. By including nonperturbative effects in the path integral, which computes coarse-grained ensemble averages, we find that the complexity saturates at late times. The saturation value is given by the exponential of the Bekenstein-Hawking entropy plus logarithmic corrections. Our results introduce a new way to compute the Hilbert space dimension of complex interacting systems from the saturating value of spread complexity.

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Phys. Rev. D 114, L081901– Published 5 October, 2026
  • Received 5 May 2026
  • Accepted 6 August 2026
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