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Exact Gravity Duals for Simple Quantum Circuits

Johanna Erdmenger, Mario Flory, Marius Gerbershagen, Michal P. Heller, Anna-Lena Weigel

TL;DR

The paper develops exact gravity duals for simple quantum circuits implementing local conformal transformations in AdS$_3$/CFT$_2$, by identifying the circuit parameter with physical boundary time and using the Fefferman–Graham expansion to map boundary data to a bulk geometry (Banados geometries). It analyzes two circuit realizations and derives the corresponding bulk metrics, providing a principled link between boundary circuit costs and bulk gravity. The authors compare holographic complexity proposals, showing that the squared Fubini–Study cost and the complexity=volume measure agree up to third order in perturbations but diverge at fourth order, indicating these costs are not universally equivalent. The framework offers a systematic route to derive bulk duals for other boundary cost functionals and to explore the gravity–complexity correspondence from first principles.

Abstract

Holographic complexity proposals have sparked interest in quantifying the cost of state preparation in quantum field theories and its possible dual gravitational manifestations. The most basic ingredient in defining complexity is the notion of a class of circuits that, when acting on a given reference state, all produce a desired target state. In the present work we build on studies of circuits performing local conformal transformations in general two-dimensional conformal field theories and construct the exact gravity dual to such circuits. In our approach to holographic complexity, the gravity dual to the optimal circuit is the one that minimizes an externally chosen cost assigned to each circuit. Our results provide a basis for studying exact gravity duals to circuit costs from first principles.

Exact Gravity Duals for Simple Quantum Circuits

TL;DR

The paper develops exact gravity duals for simple quantum circuits implementing local conformal transformations in AdS/CFT, by identifying the circuit parameter with physical boundary time and using the Fefferman–Graham expansion to map boundary data to a bulk geometry (Banados geometries). It analyzes two circuit realizations and derives the corresponding bulk metrics, providing a principled link between boundary circuit costs and bulk gravity. The authors compare holographic complexity proposals, showing that the squared Fubini–Study cost and the complexity=volume measure agree up to third order in perturbations but diverge at fourth order, indicating these costs are not universally equivalent. The framework offers a systematic route to derive bulk duals for other boundary cost functionals and to explore the gravity–complexity correspondence from first principles.

Abstract

Holographic complexity proposals have sparked interest in quantifying the cost of state preparation in quantum field theories and its possible dual gravitational manifestations. The most basic ingredient in defining complexity is the notion of a class of circuits that, when acting on a given reference state, all produce a desired target state. In the present work we build on studies of circuits performing local conformal transformations in general two-dimensional conformal field theories and construct the exact gravity dual to such circuits. In our approach to holographic complexity, the gravity dual to the optimal circuit is the one that minimizes an externally chosen cost assigned to each circuit. Our results provide a basis for studying exact gravity duals to circuit costs from first principles.
Paper Structure (18 sections, 89 equations, 2 figures)

This paper contains 18 sections, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A simple quantum circuit
  3. Generating the same sequence of states using sources
  4. General discussion
  5. Realizing case (b)
  6. Another look at the circuit from case (a)
  7. Mapping to gravity
  8. Lessons for holographic complexity
  9. Summary and outlook
  10. Acknowledgements
  11. Maximal volume slices
  12. First and second order:
  13. Third order:
  14. Fourth order:
  15. Fubini-Study complexity
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Depiction of the two circuits we consider. In case (a), the circuit evolution proceeds through a sequence of states living on time slices of different spacetimes (marked in red). There is no associated evolution with respect to physical time $t$. In case (b), the states live on different time slices of the same spacetime. In holography, in case (a) we have a sequence of independent gravity dual geometries, whereas in case (b) we arrive at a single gravity dual geometry.
  • Figure 2: Flat cylinder in which the conformal field theory lives. Black curves correspond to slices of constant time (vertical) and angle (horizontal) associated with the $w$, $\bar{w}$ coordinates. The red curves represent constant time and angle associated with the $z$, $\bar{z}$ coordinates with the now infinitesimal diffeomorphism (\ref{['eq:w-diffeo']}-\ref{['eq.barwequalsz']}) specified by $f(t,z)=z+\varepsilon (3 t^2 - 2 t^3)\sin(z)+\mathcal{O}(\varepsilon^2)$ with $\varepsilon = 0.2$ (see also \ref{['eq.confperturbative']} for a definition of infinitesimal conformal transformations).