  Here we present two novel contributions for achieving quantum advantage in
solving difficult optimisation problems, both in theory and foreseeable
practice. (1) We introduce the ''Quantum Tree Generator'', an approach to
generate in superposition all feasible solutions of a given instance, yielding
together with amplitude amplification the optimal solutions for
$0$-$1$-Knapsack problems. The QTG offers exponential memory savings and
enables competitive runtimes compared to the state-of-the-art Knapsack solver
COMBO for instances involving as few as 600 variables. (2) By introducing a
high-level simulation strategy that exploits logging data from COMBO, we can
predict the runtime of our method way beyond the range of existing quantum
platforms and simulators, for various benchmark instances with up to 1600
variables. Combining both of these innovations, we demonstrate the QTG's
potential advantage for large-scale problems, indicating an effective approach
for combinatorial optimisation problems.
