Which type of plant absorbs the most net carbon dioxide per year per square km?
I have no personal expertise in this matter, but performed a quick search of recent literature. From what I can tell, the interest in fast-growing plants is motivated primarily by the desire to maximize bioenergy production, not as a means of carbon sequestration. The typical goal is therefore multiple harvests per year in favorable conditions, meaning any carbon removed from the atmosphere by these plants is returned back to the atmosphere in a matter of months. Two plants commonly mentioned are eucalyptus trees and elephant grass.
Humbertode Jesus Eufrade Junior, et al., &quot;Sustainable use of eucalypt biomass grown on short rotation coppice for bioenergy&quot;, Biomass and Bioenergy, Vol. 90, July 2016, pp. 15-21

According to the results obtained, the density planting and fertilization levels have a greater influence on the dry matter yield, energy yield, and nutrient exports. The higher density planting reaches mean values of 30.9 tonnes of dry matter per hectare

Rafael Fiusa de Morais, et al., &quot;Elephant grass genotypes for bioenergy production by direct biomass combustion&quot;, Pesq. agropec. bras., Vol. 44, No.2, Feb. 2009

The objective of this work was to evaluate elephant grass (Pennisetum purpureum Schum.) genotypes for bioenergy production by direct biomass combustion. Five elephant grass genotypes grown in two different soil types, both of low fertility, were evaluated. [...] The dry matter yields ranged from 45 to 67 Mg ha-1

There are 100 hectares in a square kilometer, and one Mg is one metric ton, so the reported annual yields correspond to 3090 tons per square kilometer for eucalyptus, and up to 6700 tons per square kilometer for elephant grass. Obviously the dry mass of plants is not just composed of carbon. The FAO provides this helpful data:

The carbon content of vegetation is surprisingly constant across a wide variety of tissue types and species. Schlesinger (1991) noted that C content of biomass is almost always found to be between 45 and 50% (by oven-dry mass).

Assuming 45% carbon content, this means eucalyptus trees can bind 1390 tons of carbon / km2 / year and elephant grass can bind 3015 tons of carbon / km2 / year. A CO2 molecule has atomic weight 44 while carbon has atomic weight 12, so the CO2 tonnage is the 3.66-fold of these numbers, that is ~5100 tons and ~11000 tons, respectively.