The performance of the quantum adiabatic algorithm on spike Hamiltonians

• Main Authors: Kong, Linghang, Crosson, Elizabeth
• Format: Journal Article
• Language: English
• Published: 22-11-2015
• Subjects: Physics - Quantum Physics
• DOI: 10.48550/arxiv.1511.06991

Perturbed Hamming weight problems serve as examples of optimization instances
for which the adiabatic algorithm provably out performs classical simulated
annealing. In this work we study the efficiency of the adiabatic algorithm for
solving the "the Hamming weight with a spike" problem by using several methods
to compute the scaling of the spectral gap at the critical point, which apply
for various ranges of the height and width of the barrier. Our main result is a
rigorous polynomial lower bound on the minimum spectral gap for the adiabatic
evolution when the bit-symmetric cost function has a thin but polynomially high
barrier. This is accomplished by the use of a variational argument with an
improved ansatz for the ground state, along with a comparison to the spectrum
of the system when no spike term is present. We also give a more detailed
treatment of the spin coherent path-integral instanton method which was used by
Farhi, Goldstone, and Gutmann in arXiv:quant-ph/0201031, and consider its
applicability for estimating the gap for different scalings of barrier height
and width. We adapt the discrete WKB method for an abruptly changing potential,
and apply it to the construction of approximate wave functions which can be
used to estimate the gap. Finally, the improved ansatz for the ground state
leads to a method for predicting the location of avoided crossings in the
excited states of the energy spectrum of the thin spike Hamiltonian, and we use
a recursion relation to determine the ordering of some of these avoided
crossings, which may be a useful step towards understanding the diabatic
cascade phenomenon which occurs in spike Hamiltonians.