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Effect of local minima on adiabatic quantum optimization.

¹D-Wave Systems Inc., 100-4401 Still Creek Drive, Burnaby, British Columbia, V5C 6G9, Canada.

Physical Review Letters

|June 4, 2008

Summary

Adiabatic quantum computation faces challenges with problems having many local minima. The spectral gap shrinks exponentially, increasing computation time and limiting quantum advantage for these specific problem types.

Area of Science:

Quantum Computing
Computational Complexity
Quantum Optimization

Background:

Adiabatic quantum computation (AQC) is a promising paradigm for solving complex problems.
Estimating the spectral gap is crucial for determining AQC performance.
The energy landscape of a problem Hamiltonian dictates the computation's feasibility.

Purpose of the Study:

To develop a perturbative method for estimating the spectral gap in AQC.
To analyze the impact of problem Hamiltonian structure on spectral gap size.
To identify problem classes unsuitable for AQC.

Main Methods:

Perturbative analysis of energy level structures.
Focus on Hamiltonians with numerous local minima near the global minimum.

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Evaluation of spectral gap scaling with problem size.

Main Results:

A method to estimate the spectral gap based on Hamiltonian energy levels.
Demonstration that problems with many local minima lead to exponentially small spectral gaps.
Identification of a correlation between small spectral gaps and exponentially long computation times.

Conclusions:

Problems with numerous local minima near the global minimum are not suitable for standard AQC.
Quantum advantage may only be accessible via local adiabatic evolution, requiring strict phase coherence.
Understanding the energy spectrum is key to assessing AQC applicability.