Complexity: Between Field Theory and Gravity





Between Field Theory and Gravity


Quantum complexity is a fundamental concept that measures the difficulty of constructing a quantum state from a reference state and a given set of resources. Initially studied in the context of quantum computation, it has recently proven to be a powerful tool in gravitational physics, especially in quantifying the information processing capacity of black holes. The upcoming workshop aims to bring together leading experts in the field to discuss recent progress in defining complexity in theories with infinitely many degrees of freedom, and to explore the potential of quantum complexity in shedding light on the emergence of spacetime in holographic theories of quantum gravity. Through focused discussions and exchange of ideas, we hope to initiate new collaborations and research projects that will advance our understanding of this exciting and rapidly evolving field.


  • Roberto Auzzi (UCSC Brescia and INFN Perugia)

  • Elena Caceres (U. Texas, Austin)

  • Ben Craps (Vrije U. Brussels)

  • Mario Flory (Jagiellonian U. Krakov) 

  • Antonia Frassino (U. Barcelona)

  • Michal Heller (Gent U.)

  • Juan Hernandez (Vrije U. Brussels)

  • Arnab Kundu (Saha Inst. Kolkata)

  • Rene Meyer (Wurzburg U.)

  • Andrew Rolph (Amsterdam U.)

  • Adrián Sánchez-Garrido (Geneva U.)

  • Andrew Svesko (U. College London) 



  • Jose L. F. Barbon (IFT)

  • Juan F. Pedraza (IFT)

  • Monica Vergel (IFT)



This workshop is supported by:


Complexity: Between Field Theory and Gravity
  • Adrian Sanchez Garrido
  • Andrew Rolph
  • Andrew Svesko
  • Antonia Micol Frassino
  • Arnab Kundu
  • Ayan Kumar Patra
  • Ben Craps
  • Camilo las Heras
  • David Rodriguez
  • Eduardo Velasco Aja
  • Elena Caceres
  • Hyun-Sik Jeong
  • Inés Hernan
  • Jesús Huertas
  • Jose Barbon
  • Juan F. Pedraza
  • Juan Hernandez
  • Mario Flory
  • Pawel Caputa
  • Pedro Tarancón Álvarez
  • Prasoon Chakraborty
  • Rafael Carrasco Carmona
  • Rene Meyer
  • Roberto Auzzi
  • Roberto Ruiz
  • Romina Ballesteros
    • 10:30 11:30
      Integrability and complexity in quantum spin chains 1h

      There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.

      Speaker: Ben Craps
    • 11:30 12:00
      coffee break 30m
    • 12:00 13:00
      Quantum Black Holes and Holographic Complexity 1h

      In this talk, I will explore how quantum effects impact certain black holes and take into account their gravitational backreaction. Specifically, I will delve into the description of a quantum BTZ black hole (quBTZ). My discussion will include an analysis of the thermodynamic properties of these black holes. Additionally, I will examine the various complexity proposals for the quBTZ and reveal that Action Complexity fails to account for the supplementary quantum contributions and does not result in the correct classical limit. Conversely, Volume Complexity allows for a consistent quantum expansion and aligns with established boundaries.

      Speaker: Antonia Frassino
    • 13:00 15:00
      Lunch 2h
    • 15:00 16:00
      Spacetime as a quantum circuit? 1h

      We propose finite cutoff regions of holographic spacetimes as mapping between boundary states at different times and Wilsonian cutoffs. If the complexity of those quantum circuits is given by the gravitational action, we are led to equations of motion for the cutoff surface that obtain a particular geometric form, where solutions are given in terms of surfaces with constant scalar curvature. This in turn suggests a connection to the kinematic space program.

      Speaker: Mario Flory
    • 16:00 16:30
      coffee break 30m
    • 16:30 17:30
      Cost of holographic path integrals 1h

      In this talk, I will introduce holographic proposals for computational cost. If complexity is the length of the shortest path between two states, then cost is the length of a general, not-necessarily-shortest path. To highlight the differences from the holographic state complexity programme: (1) the boundary dual is cost, the "off-shell" version of complexity, (2) we consider all functions on all bulk subregions of any co-dimension (which satisfy the physical properties of cost), and (3) the proposals are by construction UV-finite. Lastly, I will explain how the path integrals, which we are proposing the cost of, fit within the framework of holographic $T\bar T$.

      Speaker: Andrew Rolph
    • 10:30 11:30
      A bulk manifestation of Krylov complexity 1h

      The double-scaled SYK (DSSYK) model provides an analytically tractable instance of low-dimensional holography in which the existence of an expansion in terms of chord diagrams allows for an explicit construction of both the bulk Hilbert space and the Hamiltonian of JT gravity from the boundary theory. In this talk, I will present recent results that build up on this framework and propose a precise entry for Krylov complexity in the holographic dictionary. I will show that the eigenstates of the Krylov complexity operator are identified with the fixed chord number states that build up the effective Hilbert space of DSSYK, therefore providing bulk length eigenstates once they are subject to the bulk/boundary map. This allows to compute the profile of bulk length as a function of time in terms of the boundary K-complexity. For the sake of completeness, a concise review on the necessary notions of K-complexity will also be provided.

      Speaker: Adrián Sánchez Garrido
    • 11:30 12:00
      coffee break 30m
    • 12:00 13:00
      State dependence of Krylov complexity in 2d CFT 1h

      We compute the Krylov Complexity of a light operator O_L in an eigenstate of a 2d CFT at large central charge. The eigenstate corresponds to a primary operator O_H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of O_H is below or above the critical dimension h_H=c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS_3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states. We will also offer brief comments on ongoing works based on the above study.

      Speaker: Arnab Kundu
    • 13:00 15:00
      Lunch 2h
    • 15:00 16:00
      Volume complexity of dS bubbles 1h

      Holographic volume complexity growth in dS stretched horizon holography has a hyperfast behaviour, which leads to a divergence in a finite time. This is very different from AdS, where the complexity rate approaches a constant value. I'll discuss holographic volume complexity in a class of asymptotically AdS geometries with dS bubbles in their interior. With the exception of the static bubble case, the complexity obtained from the volume of the smooth extremal surfaces which are anchored just to the AdS boundary has a similar behaviour to the AdS case, because it asymptotically grows linearly with time. The static bubble configuration has a zero complexity rate and corresponds to a discontinuous behaviour, which resembles a first order phase transition. If instead we consider extremal surfaces which are anchored at both the AdS boundary and the de Sitter stretched horizon, we find that complexity growth is hyperfast, as in the dS case.

      Speaker: Roberto Auzzi
    • 16:00 16:30
      coffee break 30m
    • 16:30 17:30
      Large N Qudit Complexity and Symmetry Resolution in AdS3 1h

      The fact black holes carry statistical entropy proportional to their horizon area implies that quantum information concepts are geometrized in gravity. This idea obtains a particular manifestation in the AdS/CFT correspondence, where it is believed that the quantum information content in the dual field theory state can be used to reconstruct the bulk space-time geometry. The calculation of entanglement entropy from geodesics in the bulk space-time has clarified this idea to some extent.

      In this talk, I will consider two aspects of quantum information theory in relation to holography:

      First, I will discuss the large N limit of Nielsen's operator complexity on the SU(N) manifold, with a particular choice of cost function based on the Laplacian on the Lie algebra, which leads to a polynomial (instead of exponential) penalty factors. I will first present numerical results that hint at the existence of chaotic and hence ergodic geodesic motion on the group manifold, as well show the existence of conjugate points. I will then discuss a mapping between the Euler-Arnold equation which governs the geodesic evolution, to the Euler equation of two-dimensional idea hydrodynamics, in the strict large N limit.

      Second, I will discuss a refinement of entanglement entropy for systems with conserved charges, the so-called symmetry-resolved entanglement. It measures the entanglement in a sector of fixed charge. I will present how to calculate the symmetry-resolved entanglement in two-dimensional conformal field theories with Kac-Moody symmetry, and also within W_3 higher spin theory. I will also discuss the geometric realization in the dual AdS space-time, as well as recent results including the full expansion in the UV cutoff. I will close with some comments on how symmetry resolution could be applied to quantum computational complexity.

      Speaker: René Meyer
    • 10:30 11:30
      Holographic Complexity Beyond Proposals and AdS Holography 1h
      Speaker: Michal Heller
    • 11:30 12:00
      coffee break 30m
    • 12:00 13:00
      Gravitation from optimized computation 1h

      Inspired by the universality of computation, I advocate for a notion of spacetime complexity, where gravity arises as a consequence of spacetime optimizing the computational cost of its own quantum dynamics. This principle is realized in the context of holography, where complexity is understood in terms of state preparation via Euclidean path integrals, and the linearized equations of motion for any theory of gravity emerge from the first law of complexity. This suggests gravity has a computational origin. When semi-classical bulk quantum corrections are included, the holographic first law is modified by an additional term which could be interpreted as `bulk complexity’. This leads to a derivation of semi-classical gravitational equations of motion.

      Speaker: Andrew Svesko
    • 13:00 15:00
      Lunch 2h
    • 15:00 16:00
      Lorentzian threads for generalized complexity 1h

      It was recently shown that the maximal-volume prescription for calculating holographic complexity (CV) is just one out of an infinite class of observables that display the required behavior to be the gravitational duals of complexity. This approach, later extended to include complexity-action, has enlarged and changed our understanding of holographic complexity. Many questions arise in this new framework. In this talk, I will present work in progress to understand how this new class of generalized complexities is realized in the language of Lorentzian threads.

      Speaker: Elena Cáceres
    • 16:00 16:30
      coffee break 30m
    • 16:30 17:30
      Holographic entanglement entropy and complexity in the context of braneworld models 1h

      We overview some lessons about entanglement entropy and holographic complexity in various braneworld models. In particular, we focus on the easy island model, interpreting the leading contribution of the entanglement entropy and complexity in the island phase. We also look at black holes with end-of-the world branes behind the horizon, and consider consequences of the Lloyd bound on complexity as well as the entanglement velocity bound in these models.

      Speaker: Juan Hernandez
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