Complexity and the Hilbert space dimension of 3D gravity

• URL: http://arxiv.org/abs/2602.02645v1
• Date: Mon, 02 Feb 2026 19:00:00 GMT
• Title: Complexity and the Hilbert space dimension of 3D gravity
• Authors: Vijay Balasubramanian, Rathindra Nath Das, Johanna Erdmenger, Jonathan Karl, Herman Verlinde,
• Abstract summary: We calculate the Hilbert space dimension of a black hole in 2+1-dimensional Anti-de Sitter space.<n>We find that the complexity saturates at late times.<n>Our results introduce a new way to compute the Hilbert space dimension of complex interacting systems.
• Score: 1.631115063641726
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $SL(2,\mathbb{R})$ group. By including non-perturbative 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. 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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