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Quantum adiabatic optimization and combinatorial landscapes.

V N Smelyanskiy1, S Knysh, R D Morris

  • 1NASA Ames Research Center, MS 269-3, Moffett Field, California 94035-1000, USA. Vadim.N.Smelyanskiy@nasa.gov

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 5, 2004
PubMed
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The Quantum Adiabatic Evolution algorithm faces a dynamic threshold in solving random graph satisfiability problems. Beyond this threshold, finding solutions becomes exponentially harder, impacting algorithm performance.

Area of Science:

  • Quantum computing
  • Statistical mechanics
  • Computer science

Background:

  • The satisfiability problem is a fundamental challenge in computer science.
  • Quantum Adiabatic Evolution (QAE) is a quantum algorithm with potential for solving complex problems.
  • Analyzing QAE performance on random graph instances is crucial for understanding its scalability.

Purpose of the Study:

  • To analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of the satisfiability problem.
  • To investigate the impact of the clause-to-variable ratio (gamma) on algorithm efficiency.
  • To identify and characterize a dynamic threshold affecting solution times.

Main Methods:

  • Formulating the problem as a statistical mechanics problem, focusing on the smallest eigenvalue and excitation gap.

Related Experiment Videos

  • Employing an annealing approximation with a refinement using macroscopic variables.
  • Introducing macroscopic parameters (landscapes) and a universality ansatz for random bit flips.

    • Main Results:

    • Demonstrated the existence of a dynamic threshold (gamma_d) for the QAE algorithm.
    • Showed that beyond this threshold, the algorithm's runtime increases exponentially.
    • Obtained tight upper bounds on the satisfiability transition by mapping random graphs to ensembles with reduced fluctuations.

    Conclusions:

    • The study identifies a critical threshold impacting the efficiency of Quantum Adiabatic Evolution for satisfiability problems.
    • The findings suggest limitations on QAE's practical applicability for certain problem instances.
    • The developed methods provide new insights into quantum algorithm performance and statistical mechanics of random systems.