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Polynomial-Time Classical Simulation of Quantum Ferromagnets.

Sergey Bravyi1, David Gosset1

  • 1IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA.

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Summary
This summary is machine-generated.

This study introduces a new randomized algorithm for approximating quantum spin systems. The algorithm efficiently estimates partition functions, free energy, and ground energy for models like the XY and Ising models.

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Area of Science:

  • Quantum physics
  • Statistical mechanics
  • Computational complexity

Background:

  • Quantum spin systems, including ferromagnetic XY and Ising models, are fundamental in condensed matter physics.
  • Efficiently computing their properties, such as partition functions and energies, is crucial but often computationally intractable.
  • Existing methods struggle with large system sizes and varying temperatures.

Purpose of the Study:

  • To develop a polynomial-time classical randomized algorithm for approximating the partition function of a broad family of quantum spin systems.
  • To enable efficient approximation of free energy and ground energy for these systems.
  • To leverage a special graph structure for computational tractability.

Main Methods:

  • Approximation of the partition function by a perfect matching sum on a specially constructed finite graph.
  • Utilizing a randomized algorithm based on Jerrum and Sinclair's work for efficient approximation.
  • Analysis of runtime complexity in terms of relative error (ε), system size, and inverse temperature.

Main Results:

  • A polynomial-time randomized algorithm is presented for approximating the partition function of ferromagnetic XY and Ising models.
  • The algorithm achieves a given relative error ε with runtime polynomial in ε⁻¹, system size, and inverse temperature.
  • This leads to polynomial-time algorithms for approximating free energy and ground energy to a specified additive error.

Conclusions:

  • The developed algorithm provides a significant advancement in the computational study of quantum spin systems.
  • Efficient approximation of partition functions and energies is now feasible for a wide range of models.
  • The findings have implications for understanding complex magnetic phenomena and developing new quantum materials.