Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.02645

Complexity and the Hilbert space dimension of 3D gravity

Vijay Balasubramanian, Rathindra Nath Das, Johanna Erdmenger, Jonathan Karl, Herman Verlinde

hep-thgr-qcmath-phquant-phSubmitted Feb 2, 2026• Announced Feb 4, 2026
arXivHTMLPDF

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.

Comments
11 pages

Related papers

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

Arpan Bhattacharyya, Sounak Pal, Juan F. Pedraza

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.

hep-thCross-listedarXiv:2608.19373•Aug 21, 2026

The quadratic growth of Krylov spread complexity in the BTZ black hole

Aranya Bhattacharya, Mario Flory, Michal P. Heller +2 more

The boundary quantity that captures the growth of black-hole interiors quantified by holographic complexity remains unknown beyond 2d dilaton gravity. We provide a critical analysis of a partition-function construction of Krylov spread complexity for thermofield-double states that provides a dimension-independent boundary reconstruction from semiclassical holographic partition functions, while developing the present dynamical and bulk construction for the BTZ saddle. In the double-scaled Sachdev-Ye-Kitaev model, where exact and semiclassical results can be compared, we show that the classical limit is reliable only when taken after the complexity has been reconstructed; taking this limit at the level of individual Lanczos coefficients discards essential information. Applying the construction to a large-central-charge two-dimensional conformal field theory above the Hawking-Page temperature dual to a Bañados-Teitelboim-Zanelli black hole, we find an intermediate departure from early-time quadratic growth followed by behavior compatible with a return toward asymptotically quadratic growth, rather than the linear late-time behavior of the volume and the standard finite-functional complexity = anything class. We then match this boundary behavior to a generalized complexity = anything bulk object built from an infinite series of extrinsic-curvature invariants. The construction provides a systematic route from black-hole thermodynamics to Krylov dynamics and can naturally be extended to higher-dimensional holographic black holes.

hep-thCross-listedarXiv:2608.09922•Aug 11, 2026

Variations on a Theme of Krylov

Vijay Balasubramanian, Pawel Caputa, Joan Simón

Spread complexity uses the distribution of support of a time-evolving state in the Krylov basis to quantify dispersal across accessible dimensions of a Hilbert space. Here, we describe how variations in initial conditions, the Hamiltonian, and the dimension of the Hilbert space affect spread complexity and Krylov basis structure. We introduce Koherence, the entropy of coherence between perturbed and unperturbed Krylov bases, which can, e.g., quantify dynamical amplification of differences in initial conditions in chaos. To illustrate, we show that dynamics on SL(2,R), SU(2), and Heisenberg-Weyl group manifolds, often used as paradigmatic settings for contrasting chaotic and integrable (semi-)classical behavior, display distinctively different responses to variations of the initial state or Hamiltonian. We then describe a lattice model that displays linear growth of spread complexity, saturating for bounded lattices and continuing forever in a thermodynamic limit. The latter example illustrates a breakdown of continuum/classical effective descriptions of complexity growth in bounded quantum systems.

hep-thCross-listedarXiv:2511.03775•Journal of High Energy Physics•Nov 7, 2025•7 citations

Krylov localization and suppression of complexity

Eliezer Rabinovici, Adrián Sánchez-Garrido, Ruth Shir +1 more

A bstract Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.

arXiv:2112.12128•Journal of High Energy Physics•Mar 1, 2022•123 citations

Krylov complexity in the Schrödinger field theory

Peng-Zhang He, Hai-Qing Zhang

A bstract We investigate the Krylov complexity of Schrödinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble which includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator [ H , ·] on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients { b n } are almost independent of the chemical potential, while { a n } are dependent on the chemical potential. Both { a n } and { b n } show linear relationships with respect to n . For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions | φ 0 ( t )| 2 . The Krylov complexity grows exponentially with time, but its asymptotic scaling factor λ K is significantly smaller than the twice of the slope of the { b n } coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of { b n }.

arXiv:2411.16302•Journal of High Energy Physics•Mar 19, 2025•4 citations

Cosmological Entanglement Entropy from the von Neumann Algebra of Double-Scaled SYK & Its Connection with Krylov Complexity

Sergio E. Aguilar-Gutierrez

We investigate entanglement entropy between the pair of type II$_1$ algebras of the double-scaled SYK (DSSYK) model given a chord state, its holographic interpretation as generalized horizon entropy; particularly in the (anti-)de Sitter ((A)dS) space limits of the bulk dual; and its connection with Krylov complexity. The density matrices in this formalism are operators in the algebras, which are specified by the choice of global state; and there exists a trace to evaluate their von Neumann entropy since the algebras are commutants of each other, which leads to a notion of algebraic entanglement entropy. We match it in triple-scaling limits to an area computed through a Ryu-Takayanagi formula in (A)dS$_2$ space with entangling surfaces at the asymptotic timelike or spacelike boundaries respectively; providing a first-principles example of holographic entanglement entropy for (A)dS$_2$ space. This result reproduces the Bekenstein-Hawking and Gibbons-Hawking entropy formulas for specific entangling regions points, while it decreases for others. This construction does not display some of the puzzling features in dS holography. The entanglement entropy remains real-valued since the theory is unitary, and it depends on the Krylov spread complexity of the Hartle-Hawking state. At last, we discuss higher dimensional extensions.

hep-thCross-listedarXiv:2511.03779•Journal of High Energy Physics•Nov 7, 2025•3 citations
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)SL(2,\mathbb{R})SL(2,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.
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 AdS3_33​. 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 boundary quantity that captures the growth of black-hole interiors quantified by holographic complexity remains unknown beyond 2d dilaton gravity. We provide a critical analysis of a partition-function construction of Krylov spread complexity for thermofield-double states that provides a dimension-independent boundary reconstruction from semiclassical holographic partition functions, while developing the present dynamical and bulk construction for the BTZ saddle. In the double-scaled Sachdev-Ye-Kitaev model, where exact and semiclassical results can be compared, we show that the classical limit is reliable only when taken after the complexity has been reconstructed; taking this limit at the level of individual Lanczos coefficients discards essential information. Applying the construction to a large-central-charge two-dimensional conformal field theory above the Hawking-Page temperature dual to a Bañados-Teitelboim-Zanelli black hole, we find an intermediate departure from early-time quadratic growth followed by behavior compatible with a return toward asymptotically quadratic growth, rather than the linear late-time behavior of the volume and the standard finite-functional complexity = anything class. We then match this boundary behavior to a generalized complexity = anything bulk object built from an infinite series of extrinsic-curvature invariants. The construction provides a systematic route from black-hole thermodynamics to Krylov dynamics and can naturally be extended to higher-dimensional holographic black holes.
Spread complexity uses the distribution of support of a time-evolving state in the Krylov basis to quantify dispersal across accessible dimensions of a Hilbert space. Here, we describe how variations in initial conditions, the Hamiltonian, and the dimension of the Hilbert space affect spread complexity and Krylov basis structure. We introduce Koherence, the entropy of coherence between perturbed and unperturbed Krylov bases, which can, e.g., quantify dynamical amplification of differences in initial conditions in chaos. To illustrate, we show that dynamics on SL(2,R), SU(2), and Heisenberg-Weyl group manifolds, often used as paradigmatic settings for contrasting chaotic and integrable (semi-)classical behavior, display distinctively different responses to variations of the initial state or Hamiltonian. We then describe a lattice model that displays linear growth of spread complexity, saturating for bounded lattices and continuing forever in a thermodynamic limit. The latter example illustrates a breakdown of continuum/classical effective descriptions of complexity growth in bounded quantum systems.
A bstract Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.
A bstract We investigate the Krylov complexity of Schrödinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble which includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator [ H , ·] on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients { b n } are almost independent of the chemical potential, while { a n } are dependent on the chemical potential. Both { a n } and { b n } show linear relationships with respect to n . For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions | φ 0 ( t )| 2 . The Krylov complexity grows exponentially with time, but its asymptotic scaling factor λ K is significantly smaller than the twice of the slope of the { b n } coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of { b n }.
We investigate entanglement entropy between the pair of type II1_11​ algebras of the double-scaled SYK (DSSYK) model given a chord state, its holographic interpretation as generalized horizon entropy; particularly in the (anti-)de Sitter ((A)dS) space limits of the bulk dual; and its connection with Krylov complexity. The density matrices in this formalism are operators in the algebras, which are specified by the choice of global state; and there exists a trace to evaluate their von Neumann entropy since the algebras are commutants of each other, which leads to a notion of algebraic entanglement entropy. We match it in triple-scaling limits to an area computed through a Ryu-Takayanagi formula in (A)dS2_22​ space with entangling surfaces at the asymptotic timelike or spacelike boundaries respectively; providing a first-principles example of holographic entanglement entropy for (A)dS2_22​ space. This result reproduces the Bekenstein-Hawking and Gibbons-Hawking entropy formulas for specific entangling regions points, while it decreases for others. This construction does not display some of the puzzling features in dS holography. The entanglement entropy remains real-valued since the theory is unitary, and it depends on the Krylov spread complexity of the Hartle-Hawking state. At last, we discuss higher dimensional extensions.