Time evolution of complexity: a critique of three methods

  • Tibra Ali
  • Arpan Bhattacharyya
  • S. Shajidul HaqueEmail author
  • Eugene H. Kim
  • Nathan Moynihan
Open Access
Regular Article - Theoretical Physics


In this work, we propose a testing procedure to distinguish between the different approaches for computing complexity. Our test does not require a direct comparison between the approaches and thus avoids the issue of choice of gates, basis, etc. The proposed testing procedure employs the information-theoretic measures Loschmidt echo and Fidelity; the idea is to investigate the sensitivity of the complexity (derived from the different approaches) to the evolution of states. We discover that only circuit complexity obtained directly from the wave function is sensitive to time evolution, leaving us to claim that it surpasses the other approaches. We also demonstrate that circuit complexity displays a universal behaviour — the complexity is proportional to the number of distinct Hamiltonian evolutions that act on a reference state. Due to this fact, for a given number of Hamiltonians, we can always find the combination of states that provides the maximum complexity; consequently, other combinations involving a smaller number of evolutions will have less than maximum complexity and, hence, will have resources. Finally, we explore the evolution of complexity in non-local theories; we demonstrate the growth of complexity is sustained over a longer period of time as compared to a local theory.


Effective Field Theories Lattice Quantum Field Theory AdS-CFT Correspondence Black Holes in String Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2019

Authors and Affiliations

  1. 1.Perimeter InstituteWaterlooCanada
  2. 2.Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan
  3. 3.Department of PhysicsUniversity of WindsorWindsorCanada
  4. 4.The Laboratory for Quantum Gravity & Strings, Department of Mathematics & Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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