Tracking a financial index boils down to replicating its trajectory of
returns for a well-defined time span by investing in a weighted subset of the
securities included in the benchmark. Picking the optimal combination of assets
becomes a challenging NP-hard problem even for moderately large indices
consisting of dozens or hundreds of assets, thereby requiring heuristic methods
to find approximate solutions. Hybrid quantum-classical optimization with
variational gate-based quantum circuits arises as a plausible method to improve
performance of current schemes. In this work we introduce a heuristic pruning
algorithm to find weighted combinations of assets subject to cardinality
constraints. We further consider different strategies to respect such
constraints and compare the performance of relevant quantum ans\"{a}tze and
classical optimizers through numerical simulations.Comment: 24 pages, 12 figure