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B Fefferman1, M Foss-Feig1,2,3, A V Gorshkov1,3

  • 1Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland 20742, USA.

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We demonstrate that simulating Ising spin models is computationally complex. This research suggests that certain quantum systems may soon outperform classical computers for specific sampling tasks.

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Area of Science:

  • Quantum Computing
  • Computational Complexity Theory
  • Atomic, Molecular, and Optical Physics

Background:

  • Ising spin models are fundamental in physics and can be implemented in quantum systems.
  • Classical simulation of quantum systems is a key challenge in computational complexity.
  • Boson sampling has previously shown the potential for quantum computational advantage.

Purpose of the Study:

  • To investigate the computational complexity of classically sampling from Ising spin model output distributions.
  • To construct a specific Ising Hamiltonian demonstrating this complexity.
  • To classify the sampling hardness of two-qubit commuting Hamiltonians.

Main Methods:

  • Constructing a specific Ising Hamiltonian.
  • Analyzing the output probability distribution after time evolution.
  • Relating sampling complexity to the permanent of matrices with integer entries.

Main Results:

  • The study shows that sampling from the Ising model's output distribution is classically hard.
  • The probability distribution is nearly proportional to the square of the permanent of an integer matrix.
  • This work completes the sampling hardness classification for two-qubit commuting Hamiltonians.

Conclusions:

  • Efficient classical simulation of these Ising spin dynamics is unlikely.
  • Future physical Ising spin systems may achieve problem sizes intractable for classical computers.
  • Current results focus on exact sampling hardness, with approximate sampling hardness remaining an open question for quantum supremacy demonstrations.