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Toward Using Matrix-free Tensor Decompositions to Systematically Improve Approximate Tensor-Networks.

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  • 1Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York New York 10010, United States.

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This study introduces a novel, generic tensor-network decomposition method to eliminate approximation errors. Applying canonical polyadic decomposition to coupled-cluster with single and double excitation calculations yields accurate chemical energies with reduced computational cost.

Area of Science:

  • Quantum chemistry
  • Computational physics
  • Numerical analysis

Background:

  • Tensor-network contractions are crucial in quantum chemistry, particularly in methods like coupled-cluster.
  • Approximations in tensor-network contraction can lead to error propagation, limiting accuracy.
  • Existing methods often rely on error cancellation techniques, which may not always be robust.

Purpose of the Study:

  • To develop a novel, generic tensor-network decomposition approach for accurate tensor-network approximation.
  • To eliminate error propagation inherent in traditional tensor-network approximation methods.
  • To reduce the computational scaling and storage requirements of tensor-network calculations.

Main Methods:

  • Investigated a matrix-free decomposition of full tensor-networks.

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  • Replaced exact tensor contractions in the particle-particle ladder (PPL) diagram of coupled-cluster with single and double excitation (CCSD) using canonical polyadic decomposition (CPD).
  • Utilized the iterative structure of CCSD to efficiently initialize the CPD optimization.

    • Main Results:

    • The decomposition-based approach is generic and independent of specific tensor-networks or index ordering.
    • Replaced O(N^6) tensor contractions with a potentially reduced-scaling O(N^4R) optimization problem.
    • Achieved chemically relevant energy values with errors less than 1 kcal/mol using a low CP rank.

    Conclusions:

    • The proposed decomposition method effectively eliminates error propagation in tensor-network approximations.
    • Canonical polyadic decomposition offers a viable strategy for reducing computational complexity in quantum chemical calculations.
    • This approach holds promise for achieving high accuracy in electronic structure calculations with improved efficiency.