Complexity and the Hilbert space dimension of 3D gravity

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.

💡 Research Summary

The paper presents a novel method for determining the Hilbert‑space dimension of black holes in three‑dimensional anti‑de Sitter (AdS₃) gravity by employing Krylov (or spread) complexity. Starting from an eternal black‑hole thermo‑field double (TFD) state, the authors construct the Krylov basis {|Kₙ⟩} through the Lanczos algorithm, which orthogonalizes the set of time‑evolved vectors Hⁿ|ψ₀⟩. The recursion relation H|Kₙ⟩=aₙ|Kₙ⟩+bₙ|Kₙ₋₁⟩+bₙ₊₁|Kₙ₊₁⟩ defines the Lanczos coefficients aₙ and bₙ. These coefficients are extracted from moments of the return amplitude S_TFD(t)=⟨TFD(0)|TFD(t)⟩, which in turn is given by the analytically continued partition function Z(β+it)/Z(β).

In the near‑extremal regime of the BTZ black hole, the Euclidean gravitational path integral (GPI) yields a spectral density ρ(E) that, after incorporating non‑perturbative off‑shell contributions, takes the positive form ρ(E)≈A e^{S₀}(E−E₀)^{½}. Using this density, the Lanczos coefficients are found to be aₙ=−(4n+3)/(2β)−E₀ and bₙ=1/β √{n(2n+1)}. These match the coefficients for a particle moving on the SL(2,ℝ) group manifold, reflecting the underlying SL(2,ℝ) symmetry of the dimensionally reduced JT‑like sector. Consequently, the orthogonal polynomials hₙ(E) that appear in the Krylov expansion are generalized Laguerre polynomials Lₙ^{(1/2)}.

If one substitutes the continuous density directly into the spread‑complexity formula C_S(t)=∑ₙ n|ψₙ(t)|², the result grows quadratically with time (C_S∝t²) and never saturates, because the spectrum is treated as continuous with infinitely many states. The authors argue that the GPI actually computes an ensemble‑averaged quantity, smoothing over the fundamentally discrete spectrum of the full quantum gravity theory. To capture the discreteness, they include the leading non‑perturbative wormhole contribution that connects the two boundaries of the Euclidean geometry. This contribution produces the universal “sine kernel” of the Gaussian Unitary Ensemble (GUE):

ρ(E)ρ(E′)≈ρ(E)ρ(E′)−