Evidence of Performance–benefits for Quantum Adiabatic Search from Noise–induced Sampling of Alternative Hamiltonian Paths

In the original presentation of quantum adiabatic search (QuAdS) the time-dependent search Hamiltonian linearly interpolates from initial to final Hamiltonian. Farhi and coworkers have suggested (quant-ph/0208135) that QuAdS performance might benefit from evolving the search Hamiltonian along paths different from the linearly interpolating one. In this paper QuAdS is numerically simulated to study its performance in the presence of non-uniform noise in which each qubit is acted on by a different noise field. QuAdS is used to find solutions to randomly generated instances of the NP-Complete problem N-bit Exact Cover 3 which have a unique solution. The simulations determine the algorithm’s noise-averaged median runtime as a function of the number of bits N for 7 ≤ N ≤ 16, and for various values of noise-power. Power-law and exponential scaling relations are fit to the simulation results. Power-law (exponential) scaling is found to provide an excellent (good) fit to our data. The scaling exponent is found to increase at first, then peak and begin to decrease as noise power is increased. The noiseless part of our search Hamiltonian evolves along the linearly interpolating (Hamiltonian) path. Noise causes the search Hamiltonian to deviate from this path, and we argue that our results provide evidence for the Farhi et. al. suggestion that such deviations may benefit QuAdS performance. Although decoherence produced by sufficiently large noise power should ultimately rob QuAdS of its quantum performance-enhancements, our results suggest the existence of a window of noise parameter values in which noise assists QuAdS.