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First-order phase transition in the quantum adiabatic algorithm.

A P Young1, S Knysh, V N Smelyanskiy

  • 1Department of Physics, University of California, Santa Cruz, California 95064, USA.

Physical Review Letters
|April 7, 2010
PubMed
Summary
This summary is machine-generated.

We simulated the quantum adiabatic algorithm for the exact cover problem. At large problem sizes, we observed a quantum phase transition that becomes more frequent as the size increases.

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Area of Science:

  • Quantum computing
  • Computational complexity theory
  • Statistical mechanics

Background:

  • The quantum adiabatic algorithm (QAA) is a leading quantum algorithm for solving complex computational problems.
  • The exact cover problem is a classic NP-complete problem with applications in various fields.
  • Understanding the performance of QAA for hard problems is crucial for assessing its practical utility.

Purpose of the Study:

  • To investigate the behavior of the quantum adiabatic algorithm (QAA) for the exact cover problem at large scales.
  • To identify and characterize phase transitions occurring during the QAA's evolution.
  • To determine the prevalence of these transitions as a function of problem size.

Main Methods:

  • Quantum Monte Carlo simulations were employed to model the QAA.
  • Parallel tempering techniques were incorporated to enhance simulation efficiency and explore the phase space.
  • Simulations were performed for exact cover problem instances up to size N=256.

Main Results:

  • At large problem sizes (large N), a significant fraction of exact cover instances exhibit a discontinuous, first-order quantum phase transition during QAA evolution.
  • The proportion of instances displaying this phase transition increases with increasing problem size N.
  • Extrapolation suggests that this fraction may approach 1 as N tends to infinity.

Conclusions:

  • The prevalence of first-order quantum phase transitions in QAA for the exact cover problem grows with problem size.
  • These findings indicate potential challenges for the scalability and effectiveness of QAA for large instances of this problem.
  • Further research is needed to understand the implications of these transitions for quantum computation.