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  • Letter
  • Open Access

Building Krylov complexity from circuit complexity

Chenwei Lv1,*, Ren Zhang2,*, and Qi Zhou1,3,†

  • *These authors contributed equally to this paper.
  • †Contact author: zhou753@purdue.edu
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Phys. Rev. Research 6, L042001 – Published 1 October, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L042001

Abstract

Krylov complexity has emerged as a probe of operator growth in a wide range of nonequilibrium quantum dynamics. However, a fundamental issue remains in such studies: the definition of the distance between basis states in Krylov space is ambiguous. Here we show that Krylov complexity can be rigorously established from circuit complexity when dynamical symmetries exist. Whereas circuit complexity characterizes the geodesic distance in a multidimensional operator space, Krylov complexity measures the height of the final operator in a particular direction. The geometric representation of circuit complexity thus unambiguously designates the distance between basis states in Krylov space. This geometric approach also applies to time-dependent Liouvillian superoperators, where a single Krylov complexity is no longer sufficient. Multiple Krylov complexity may be exploited jointly to fully describe operator dynamics.

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Outline

Information

Phys. Rev. Research 6, L042001– Published 1 October, 2024
  • Received 4 May 2023
  • Revised 2 October 2023
  • Accepted 3 September 2024
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