Krylov complexity and the growth of the black hole interior in 3D gravity

Abstract: We investigate the growth of the black hole interior in three-dimensional gravity from the boundary theory. For the two-sided BTZ black hole, we propose a boundary reconstruction of the time dependence of a codimension-one surface in terms of correlation functions of smeared operators in the thermofield double state, reproducing the characteristic late-time linear growth predicted by the complexity-volume proposal. Using the Chern--Simons formulation of three-dimensional gravity, these nonlocal correlators are represented by bulk Wilson lines with smeared endpoints, extending the familiar connection between Wilson lines and codimension-two observables underlying holographic entanglement entropy to codimension-one observables. We then ask whether the same geometric growth is captured by Krylov complexity. For the smeared operators, we extract the Lanczos data from their correlation functions and find that operator Krylov complexity reproduces the late-time linear growth of the black hole interior, extending previous connections between operator growth and bulk geometry to AdS$_3$. By contrast, the Krylov spread complexity of the thermofield double state, obtained from the semiclassical gravitational partition function, does not exhibit the linear growth of the bulk volume within the regime accessible to our analysis. Our results therefore point to a distinguished role for operator Krylov complexity in encoding black hole interior growth beyond two-dimensional gravity, while highlighting a qualitative distinction between operator and state notions of Krylov complexity.

• .The paper demonstrates that operator Krylov complexity, using spatially smeared operators, reproduces the linear growth of the BTZ interior volume, suggesting a direct link between complexity and holographic space-time evolution.
• It identifies a strict qualitative difference between operator Krylov complexity and spread complexicity, the former aligning with the expected volume growth pattern, the latter showing quadratic rather than linear growth.
• The study underscores the dependence on the specific Krylov construction, emphasizing the importance of careful definition and analysis when extending these concepts to higher dimensions, setting forth a framework for further exploration of these ideas in 3D gravity systems.

Overview

This paper by Bhattacharyya, Pal, and Pedraza (2608.19373) addresses a concrete question in holographic quantum gravity: can the linear late-time growth of the two-sided BTZ interior, as prescribed by the Complexity$=$Volume (CV) proposal, be reconstructed from boundary observables and captured by Krylov complexity? The question is well posed because the analogous construction in AdS$_2$/JT gravity is well established: there, the codimension-one volume reduces to the length of the Einstein–Rosen bridge, which is encoded in heavy-operator two-point functions and saturates non-perturbatively via matrix-model effects (Iliesiu et al., 2021). In AdS$_3$, however, the CV surface is anchored on spatial circles rather than points, so its volume is not naturally read off from local correlators. The paper overcomes this obstacle with spatially smeared operators plus Euclidean thermal-circle averaging, verifies the construction in the Chern–Simons formulation, and then contrasts operator and state notions of Krylov complexity. Its central finding is a sharp qualitative split: operator Krylov complexity reproduces the linear interior growth, whereas Krylov spread complexity of the thermofield double (TFD) state grows only quadratically.

Boundary reconstruction of the BTZ interior

The proposed observable is an integrated, smeared TFD two-point function,

$\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$

where $\widetilde{\mathcal O}$ is a primary smeared along the spatial $S^1$ with a periodic Gaussian kernel $$K(\phi,\phi')=\mathcal N_\sigma\sum_n e^{-\sigma n^2}e^{\frac{2\pi i n}{L}(\phi-\phi')}$$ that reduces to a delta function as $\sigma\to0$, and the Euclidean integrals run over symmetric segments of the thermal circle. The temporal averaging respects KMS periodicity and is motivated by information-metric constructions (Miyaji, 2016, Miyaji et al., 2015), though the observable differs from those constructions in its spatial smearing kernel and its lack of a Fisher-information interpretation.

Bulk evaluation uses Wilson lines in infinite-dimensional representations within $SL(2,\mathbb R)\times SL(2,\mathbb R)$ Chern–Simons theory (Castro et al., 2016). Fixed-endpoint Wilson lines reproduce primary two-point functions; smearing each endpoint with $K$ after evaluating the line extends the construction to codimension-one observables. Working in Kruskal gauge on the non-rotating BTZ background and retaining only the leading Rindler image contribution at high temperature, the invariant entering the line takes the form $_2$0, with all Lorentzian time dependence isolated in the first term. For $_2$1, the Euclidean-time-integrated observable exhibits, for $_2$2,

$_2$3

with a bracketing factor involving $_2$4 and $_2$5 corrections that becomes trivially extensive upon decompactification ($_2$6). This is precisely the characteristic linear growth of the codimension-one BTZ volume. The authors are careful to note that the overall coefficient depends on normalization and smearing prescription, so the result should not be read as a universal equality of growth rates with CV — the robust feature is the shared linear time dependence. Conceptually, this suggests extending the familiar Wilson-line/holographic-entanglement-entropy dictionary (Ammon et al., 2013) to codimension-one observables associated with complexity. The analysis is explicitly semiclassical: no statement is made about the saturation expected on timescales of order $_2$7, and no JT-like non-perturbative matrix-model completion is available for AdS$_2$8 in comparable generality.

Operator Krylov complexity of smeared operators

For the spatially smeared operators, the paper extracts Lanczos data from the thermal autocorrelation function using the moment method. Expanding the smeared correlator in spatial Fourier modes yields Appell $_2$9 functions with indices $_3$0; analytic control is achieved for $_3$1 in the zero-mode sector, where the microcanonical spectral measure obtained by inverse Laplace/Fourier transforms behaves as

$_3$2

The even moments are then edge-dominated, $_3$3 with subexponential $_3$4, implying a Lanczos plateau $_3$5 and hence ballistic propagation:

$_3$6

This linear late-time regime mirrors the bulk surface growth, extending the operator-growth/geometry connection (Kar et al., 2021, Ambrosini et al., 2024) to AdS$_3$7. Two caveats are stated plainly. First, dependence on the detailed smearing profile enters through nonzero Fourier modes and was left unanalyzed, so the kernel-independence of the linear rate is established only within the zero mode. Second, compared to the JT case, where the spectral measure first produces a Lanczos ascent before reaching the plateau, the extended operators here reach the edge-dominated plateau more directly — the authors note it remains open whether this difference is attributable to non-locality, since a direct computation of matrix elements would be required. Extending the Krylov construction to include the Euclidean-time integration itself also requires care, since time-dependent integration limits do not define an autocorrelator of a fixed seed; the authors propose a fixed-filter convolution as a possible resolution and defer it.

Spread complexity fails to track the volume

The state-side analysis proceeds from the semiclassical partition function $_3$8, giving survival amplitude $_3$9 and moments expressible through Laguerre polynomials. Numerical fits over accessible Krylov indices ($\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$0) suggest coefficients of the form $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$1, $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$2. Extrapolated naively to large $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$3, these would give hyperbolic (exponential) spread complexity $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$4. The paper argues this extrapolation is inconsistent with the full moment problem: solving the exact Lanczos recursion from the semiclassical moments shows instead

$\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$5

which sits precisely at the critical point $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$6 of the effective $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$7 dynamics, i.e., $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$8, giving quadratic rather than exponential growth. Direct evolution of the chain built from the first $\int d\tau_1^E\,d\tau_2^E\;\langle \widetilde{\mathcal O}_L\,\widetilde{\mathcal O}_R\rangle_{\rm TFD}\;\sim\; \mathbfcal A^{L|R}(C),$9 Lanczos coefficients confirms approximately quadratic behavior over the accessible range. Independent diagnostics reinforce that the moment problem is not exactly $\widetilde{\mathcal O}$0: the second difference $\widetilde{\mathcal O}$1, the mismatch $\widetilde{\mathcal O}$2 at fixed asymptotic slope, a time-dependent effective representation parameter $\widetilde{\mathcal O}$3 extracted from the normalized variance, and the failure of the asymptotic solution's zeroth amplitude to reproduce $\widetilde{\mathcal O}$4 — a discrepancy traced to extending large-$\widetilde{\mathcal O}$5 coefficients down to $\widetilde{\mathcal O}$6 rather than to any inconsistency in the state choice. The conclusion is a strong negative claim: within the semiclassical analysis, spread complexity does not exhibit the linear CV-type growth, in contrast to the special two-dimensional settings where a direct correspondence exists (Rabinovici et al., 2023, Heller et al., 2024).

Limitations and open questions

Several limitations bound the results. The reconstruction is semiclassical throughout; non-perturbative saturation of both the bulk volume and the complexities is unaddressed, and the absence of a general AdS$\widetilde{\mathcal O}$7 matrix-model dual prevents a JT-style treatment. The operator-complexity analysis is analytic only for $\widetilde{\mathcal O}$8 in the zero spatial Fourier mode, with the $\widetilde{\mathcal O}$9 case verified only at the level of edge-dominated scaling; the dependence of the linear rate on the smearing width and profile is unresolved. Whether ETH-like descriptions apply to the smeared non-local operators, and whether different kernels select different codimension-one observables within the "complexity$S^1$0anything" landscape, remain open. On the state side, the strict late-time asymptotics of spread complexity is not rigorously established — the quadratic behavior persists over the numerically accessible range and is strongly suggestive but not proven, leaving open whether a linear regime could emerge at later times or whether quantum corrections alter the picture. A concurrent work (Bhattacharya et al., 10 Aug 2026), using early-time expansion with Padé continuation, reaches a consistent picture of return-to-quadratic growth while likewise not fixing strict asymptotics.

Conclusion

The paper delivers two related results. First, a spatially smeared, thermally integrated TFD correlator — represented in the bulk as an endpoint-smeared Chern–Simons Wilson line — reproduces the linear late-time growth of the codimension-one BTZ surface, providing a three-dimensional extension of the boundary-correlator picture familiar from AdS$S^1$1/JT. Second, it establishes an asymmetry between Krylov notions: operator Krylov complexity of the same smeared operators develops the characteristic linear regime via a Lanczos plateau, while the spread complexity of the TFD state, governed by an asymptotically critical $S^1$2-like but not exact chain, remains approximately quadratic and does not track the interior volume. Together these results indicate that beyond two-dimensional gravity, the relation between Krylov complexity and black hole interior growth depends sensitively on the choice of construction, with operator complexity currently holding the more direct geometric interpretation.