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Uncomputability and complexity of quantum control.

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Quantum control problems are generally not algorithmically solvable due to finite precision, even for approximate solutions. However, specific problem classes or sufficiently rich controls can ensure algorithmic solvability.

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Area of Science:

  • Quantum physics
  • Theoretical computer science
  • Computational complexity

Background:

  • Physical quantities are known with finite precision, represented by rational numbers.
  • Quantum control aims to manipulate quantum system dynamics using external interactions.
  • The algorithmic solvability of quantum control problems is a key theoretical question.

Purpose of the Study:

  • To determine the general algorithmic solvability of quantum control problems for both open and closed systems.
  • To investigate the impact of finite precision on the computability of quantum control.
  • To explore the complexity classes of quantum control problems.

Main Methods:

  • Establishing an equivalence between quantum control problems and Diophantine equations.
  • Utilizing laboratory and numerical experiments to validate theoretical deductions.
  • Analyzing the computational complexity of specific quantum control scenarios.

Main Results:

  • Quantum control problems are generally not algorithmically solvable, even for approximate targets.
  • The technique reveals connections between quantum control and polynomial equations with integer unknowns.
  • A two-mode coherent field control problem is demonstrated to be NP-hard.

Conclusions:

  • The inherent limitations of finite precision render most quantum control problems undecidable.
  • Algorithmic solvability can be achieved for specific quantum control problem classes or with extensive control sets.
  • The study challenges assumptions about the computational simplicity of two-body quantum problems.