phase space generalizations have also emerged,[@takabayasi54; @shalashilin00; @burghardt01a; @burghardt01b] of particular relevance for dissipative systems.[@trahan03b; @donoso02; @bittner02a; @hughes04]

Despite this success, QTMs suffer from a significant numerical drawback, which, to date, precludes a completely robust application of these methods. Namely: QTMs are numerically unstable in the vicinity of amplitude nodes. This “node problem” manifests in several different ways:[@wyatt01b; @wyatt] (1) infinite forces, giving rise to kinky, erratic trajectories; (2) compression/inflation of trajectories near wavefunction local extrema/nodes, leading to; (3) insufficient sampling for accurate derivative evaluations. Nodes are usefully divided into two categories,[@poirier04bohmI] depending on whether $Q$ is formally well-behaved (“type one” nodes) or singular (“type two” nodes). For stationary state solutions to the  equation, for instance, all nodes are type one nodes. In principle, type one nodes are “gentler” than type two nodes; however, from a numerical standpoint, even type one nodes will give rise to the problems listed above, because the slightest numerical error in the evaluation of $Q$ is sufficient to cause instability.

In the best case, the node problem simply results in substantially more trajectories and time steps than the corresponding classical calculation; in the worst case, the QTM calculation may fail altogether, beyond a certain point 