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Solution to the quantum Zermelo navigation problem.

Dorje C Brody1,2, David M Meier1

  • 1Department of Mathematics, Brunel University, Uxbridge UB8 3PH, United Kingdom.

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We found a simple solution for time-optimal quantum control, even with background fields. This quantum Zermelo problem solution is analogous to classical navigation, using geodesic curves on a Randers metric space.

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Area of Science:

  • Quantum mechanics
  • Optimal control theory
  • Geometric mechanics

Background:

  • Generating specific unitary gates is crucial for quantum computing.
  • Uncontrollable ambient Hamiltonians, like background fields, complicate control.
  • The classical Zermelo navigation problem addresses time-optimal steering with external forces.

Purpose of the Study:

  • To find the time-optimal control Hamiltonian for generating unitary gates in the presence of ambient fields.
  • To solve the quantum Zermelo problem and provide an explicit, simple solution.

Main Methods:

  • The study adapts the classical Zermelo navigation problem's approach.
  • It involves finding geodesic curves on the space of unitary operators equipped with a Randers metric.
  • Explicit solutions to the associated geodesic equations of motion are derived.

Main Results:

  • A remarkably simple solution to the quantum Zermelo problem is obtained.
  • The optimal control strategy is shown to effectively utilize the ambient field, akin to 'going with the wind'.

Conclusions:

  • The solution provides a method for efficient quantum gate synthesis under realistic environmental conditions.
  • The findings offer a new perspective on optimal control in quantum systems by leveraging background dynamics.