The quantum adiabatic optimization algorithm and local minima

Authors:
Ben W. Reichardt
Affiliations:
University of California Berkeley, Berkeley, CA
Venue:
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Year:
2004

Citing 2
Cited 7

The algebraic eigenvalue problem

The algebraic eigenvalue problem
How Powerful is Adiabatic Quantum Computation?

FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science

Guest Column: NP-complete problems and physical reality

ACM SIGACT News
Quantum Hypercomputability?

Minds and Machines
Minor-embedding in adiabatic quantum computation: I. The parameter setting problem

Quantum Information Processing
A novel strategy for multi-resource load balancing in agent-based systems

International Journal of Intelligent Information and Database Systems
Quantum adiabatic algorithms, small gaps, and different paths

Quantum Information & Computation
Different adiabatic quantum optimization algorithms for the NP-complete exact cover and 3SAT problems

Quantum Information & Computation
Partial adiabatic quantum search algorithm and its extensions

Quantum Information Processing

Quantified Score

Hi-index	0.01

Visualization

Abstract

The quantum adiabatic optimization algorithm uses the adiabatic theorem from quantum physics to minimize a function by interpolation between two Hamiltonians. The quantum wave function can sometimes tunnel through significant obstacles. However it can also sometimes get stuck in local minima, even for fairly simple problems. An initial Hamiltonian which insufficiently mixes computational basis states is analogous to a poorly mixing Markov transition rule. We study a physical system -- the Ising quantum chain with alternating sector interaction defects, but constant transverse field -- which is equivalent to applying the quantum adiabatic algorithm to a particular SAT problem. We prove that for a constant range of values for the transverse field, the spectral gap is exponentially small in the sector length. Indeed, we prove that there are exponentially many eigenvalues all exponentially close to the ground state energy. Applying the adiabatic theorem therefore takes exponential time, even for this simple problem.