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  • Letter
  • Open Access

Krylov complexity is not a measure of distance between states or operators

Sergio E. Aguilar-Gutierrez1,* and Andrew Rolph2,3,†

  • 1Institute for Theoretical Physics, KU Leuven, 3001 Leuven, Belgium
  • 2Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, The Netherlands
  • 3Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB) and The International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium

  • *sergio.ernesto.aguilar@gmail.com
  • †andrew.d.rolph@gmail.com
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Phys. Rev. D 109, L081701 – Published 22 April, 2024

DOI: https://doi.org/10.1103/PhysRevD.109.L081701

Abstract

We ask whether Krylov complexity is mutually compatible with the circuit and Nielsen definitions of complexity. We show that the Krylov complexities between three states fail to satisfy the triangle inequality and so cannot be a measure of distance: there is no possible metric for which Krylov complexity is the length of the shortest path to the target state or operator. We show this explicitly in the simplest example, a single qubit, and in general.

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Physics Subject Headings (PhySH)

Corrections

6 August, 2024

Correction: Support information in the second sentence in the Acknowledgements section was incomplete and has been fixed.

Article Text

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Outline

Information

Phys. Rev. D 109, L081701– Published 22 April, 2024
  • Received 23 November 2023
  • Accepted 15 March 2024
  • Corrected 6 August 2024
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