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We found a connection between Krylov complexity and Nielsen complexity, two measures of quantum evolution. The average Krylov complexity relates to a matrix that bounds Nielsen complexity.

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Area of Science:

  • Quantum Information Science
  • Quantum Computing
  • Quantum Chaos

Background:

  • Krylov complexity and Nielsen complexity quantify quantum evolution.
  • These approaches originate from quantum chaos and quantum computation, respectively.
  • Existing research has largely treated these complexities independently.

Purpose of the Study:

  • To investigate the relationship between Krylov complexity and Nielsen complexity.
  • To bridge the gap between quantum chaos and quantum computation perspectives on complexity.

Main Methods:

  • Mathematical analysis connecting Krylov complexity and Nielsen complexity.
  • Utilizing matrix properties and geodesic flow concepts.
  • Developing a custom penalty schedule for Nielsen complexity.

Main Results:

  • Demonstrated a direct relationship between Krylov and Nielsen complexities.
  • Expressed the time average of Krylov complexity as a matrix trace.
  • Established an upper bound for Nielsen complexity using this matrix.

Conclusions:

  • Krylov and Nielsen complexities are mathematically linked.
  • This connection offers new insights into quantum evolution.
  • Potential for unified understanding of quantum complexity measures.