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What can quantum optics say about computational complexity theory?

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This study explores sampling from linear-optical networks with Gaussian states, finding output probabilities relate to matrix permanents. An efficient algorithm exists for sampling thermal states, impacting quantum and computational complexity theory.

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Area of Science:

  • Quantum optics
  • Computational complexity theory

Background:

  • Sampling from linear-optical networks with Gaussian states presents theoretical challenges.
  • Understanding output probability distributions is crucial for quantum information processing.

Purpose of the Study:

  • To derive a general formula for output probabilities in linear-optical networks.
  • To analyze the computational complexity of sampling from these distributions for various input states.

Main Methods:

  • Derivation of a general formula for output probabilities.
  • Analysis of output probabilities for thermal and squeezed-vacuum states.
  • Investigation of computational complexity classes, including #P-hard problems and BPP^{NP}.

Main Results:

  • Output probabilities for thermal states are proportional to permanents of positive-semidefinite Hermitian matrices.
  • An efficient classical algorithm exists for sampling from the output probability distribution for thermal states, placing it in the BPP^{NP} complexity class.
  • The complexity of sampling for squeezed-vacuum states is also discussed.

Conclusions:

  • The research provides a method for calculating output probabilities and demonstrates an efficient sampling algorithm for specific Gaussian states.
  • The findings bridge quantum theory and computational complexity, suggesting practical implications for quantum computing and simulation.