  It has been suggested that excitation transport in photosynthetic light
harvesting complexes features speedups analogous to those found in quantum
algorithms. Here we compare the dynamics in these light harvesting systems to
the dynamics of quantum walks, in order to elucidate the limits of such quantum
speedups. For the Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria,
we show that while there is indeed speedup at short times, this is short lived
(70 fs) despite longer lived (ps) quantum coherence. Remarkably, this time
scale is independent of the details of the decoherence model. More generally,
we show that the distinguishing features of light-harvesting complexes not only
limit the extent of quantum speedup but also reduce rates of diffusive
transport. These results suggest that quantum coherent effects in biological
systems are optimized for efficiency or robustness rather than the more elusive
goal of quantum speedup.
