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Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations.

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Instantaneous quantum polynomial time (IQP) computations strengthen the idea that quantum computers are hard to simulate. This study shows IQP hardness holds if specific average-case complexity conjectures are true.

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Area of Science:

  • Quantum Computing
  • Computational Complexity Theory
  • Statistical Mechanics

Background:

  • The simulation of quantum computations by classical computers is a central challenge in quantum information science.
  • Instantaneous quantum polynomial time (IQP) computations represent a significant class of quantum tasks.
  • The hardness of IQP computations is crucial for establishing the power of quantum computers.

Purpose of the Study:

  • To strengthen the conjecture that quantum computers are classically hard to simulate using IQP computations.
  • To investigate the relationship between average-case hardness conjectures and the classical simulation of IQP.
  • To explore connections between quantum complexity and problems in statistical mechanics and algebra.

Main Methods:

  • Utilizing the class of commuting quantum computations known as IQP.
  • Formulating average-case hardness conjectures related to the Ising model and low-degree polynomials.
  • Deriving spin-based generalizations of the boson sampling problem.
  • Analyzing worst-case complexity to validate the proposed conjectures.

Main Results:

  • Demonstrating that IQP computations are classically hard to simulate up to constant additive error, contingent on the validity of two plausible average-case hardness conjectures.
  • Establishing that these conjectures hold in the setting of worst-case complexity.
  • Developing novel spin-based generalizations of boson sampling, bypassing the permanent anticoncentration conjecture.

Conclusions:

  • The study provides strong evidence for the classical hardness of IQP computations, reinforcing the quantum computing advantage.
  • The findings link quantum complexity with established problems in statistical mechanics and algebraic complexity.
  • The work offers new theoretical tools and insights into the simulation of quantum systems.