concept.

The quality of the initial state $\mathcal{S}$ is likely to influence dramatically the reverse annealing process. For our experiments we use a classical greedy algorithm to set $\mathcal{S}$, as described in the next section.\

Experimental Results {#sec:main}
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We aim to solve representative portfolio instances at the limit of programmability of the most advanced DW2000Q device, and to benchmark the results by means of a competitive heuristics running on classical computers. Note that due to the embedding overhead on DW2000Q we can embed a maximum of 64 logical binary variables on a fully-connected graph, which means that the largest search space for our benchmarks is around $60!/(30!)^2\simeq$ hundred thousand trillion if $M=N/2$. Our instance set consists of 30 randomly generated instances for each of the following number of assets $N=${24, 30, 36, 42, 48, 54, 60}. In Table \[tab:GreedySearchAndParams\] we show the expected number of assets in the optimal portfolio of our unconstrained problem sets, computed by using all methods in this study, showing that the search space is non-trivial for all tested cases. Since exhaustive search is out of question for the largest problems, we provide two algorithms below that will be used to have a reference benchmark 