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Towards a polynomial algorithm for optimal contraction sequence of tensor networks from trees.

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Area of Science:

  • Computational physics
  • Quantum information science
  • Applied mathematics

Background:

  • Tensor network contraction sequence optimization is computationally challenging, proven to be NP-complete for arbitrary networks.
  • Efficient tensor network contraction is crucial for various scientific simulations.

Purpose of the Study:

  • To investigate if computational complexity, rather than computational cost, allows for a polynomial solution to tensor network contraction.
  • To develop a polynomial algorithm for optimizing tensor contraction sequences in tensor tree networks.

Main Methods:

  • Conjecturing a polynomial complexity for optimal contraction based on computational complexity.
  • Developing and proving a polynomial algorithm for tensor tree networks.
  • Conducting numerical simulations to validate the algorithm's performance.

Main Results:

  • A polynomial algorithm for optimal contraction complexity in tensor tree networks is proposed.
  • The algorithm guarantees minimal time complexity and linear space complexity simultaneously for tensor tree networks.
  • Numerical simulations demonstrate significant performance benefits for large-scale networks.

Conclusions:

  • The proposed polynomial algorithm offers an efficient solution for tensor tree network contraction.
  • This work has potential for optimizing physical simulations and advancing tensor network computational complexity research.