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Hamiltonian Simulation with Random Inputs.

Qi Zhao1,2, You Zhou3,4, Alexander F Shaw1

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

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We introduce average-case error analysis for digital quantum simulations, showing significant error reduction compared to worst-case bounds. This new theory accurately predicts average error for Hamiltonian simulation methods.

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Area of Science:

  • Quantum Computing
  • Computational Physics
  • Quantum Information Science

Background:

  • Algorithmic error in digital quantum simulations is typically assessed using worst-case spectral norm analysis.
  • This worst-case analysis can be overly pessimistic for practical applications.

Purpose of the Study:

  • To develop a theory for the average-case performance of Hamiltonian simulation using random initial states.
  • To provide a more realistic error assessment beyond pessimistic worst-case bounds.

Main Methods:

  • Developed a theory relating average-case error to the Frobenius norm of multiplicative error.
  • Derived upper bounds for product formula (PF) and truncated Taylor series methods.
  • Applied the theory to general lattice and k-local Hamiltonians, including the Heisenberg chain.

Main Results:

  • Established bounds for average-case error in digital Hamiltonian simulation.
  • Demonstrated a quadratic reduction in error for the Heisenberg chain from O(n) (worst-case) to O(sqrt[n]) (average-case).
  • Numerical evidence supports the accuracy of the developed average-case error theory.

Conclusions:

  • Average-case analysis offers a more practical and less pessimistic view of digital quantum simulation errors.
  • The developed theory provides valuable insights for error mitigation and performance estimation in quantum algorithms.
  • Results are applicable to quantum scrambling simulations and other complex quantum systems.