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Quantum circuit complexity for qudit systems is explored using Riemannian geometry. Optimal circuits correspond to geodetic evolutions, with complexity depending on approximation error.

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Area of Science:

  • Quantum Information Science
  • Quantum Computation Theory
  • Mathematical Physics

Background:

  • Quantum circuit complexity is a fundamental problem in quantum computation.
  • Previous studies primarily focused on qubit systems.
  • Extending these concepts to qudit (quantum digit) systems is crucial for advancing quantum information processing.

Purpose of the Study:

  • To investigate quantum circuit complexity for qudit systems.
  • To establish a connection between optimal quantum circuits and geometric concepts.
  • To analyze the dependence of circuit complexity on approximation errors.

Main Methods:

  • Utilizing Riemannian geometry to model quantum system evolution.
  • Analyzing the parametrization of the SU(d(n)) group, which describes qudit systems.
  • Deriving the relationship between geodetic evolutions and optimal quantum circuits.

Main Results:

  • Optimal quantum circuits for qudit systems are shown to be equivalent to geodetic evolutions.
  • The study reveals an explicit dependence of quantum circuit complexity on controllable approximation error bounds.
  • This geometric approach provides a new perspective on quantifying quantum computational complexity.

Conclusions:

  • The geometric framework offers a powerful tool for understanding and optimizing quantum circuits in qudit systems.
  • Controlling approximation errors is key to managing the complexity of quantum computations.
  • This research contributes to the theoretical foundations of advanced quantum computing.