The carbon source of plants is CO2 carbon dioxide, which it takes from the air or dissolved in water. Plants cannot assimilate anything else. Science has long looked for replacements, and the only compound that can replace CO2 is the stuff called "EXCEL". It's a fairly simple carbon compound dissolved in water. When absorbed by the plant tissue it brakes down into CO2 + something else (forgot). It is sold at aquarium stores.

Info on EXCEL on: Seachem Flourish Excel

BTW: EXCEL is glutaraldehyde.

This is the basis of a lot of plant nutrition supplements, like AN's Carboload and others. You'll hear about a lot of people growing things they're not supposed to grow adding molasses to their water, but that's not a smart thing to do in hydroponics usually.

Adding carbs is good for most plants, but hydro systems can be touchy because they're not a full ecosystem like soil is. You provide a lot of high-energy food sources in the absence of beneficial bacteria and you can suddenly find your reservoir packed full of the bad kind.

So don't just dump sugar in your res. Get something designed for hydroponic use if you want to go that route, and consider colonizing the solution with beneficial bacteria and fungi (like you can get in AN's Piranha and Tarantula). With a good colony of beneficials the bad microbes can't really get a foothold.

Carbo loading, particularly molasses, stimulates bacterial growth. No plant can take up carbon via its roots. Its not designed that way.
But if you want to go the bacterial route you might as well go back to dirt growing. There you have the opportunity to manufacture plant nutrients from waste material by bacteria. A cheap, but inefficient system since a lot of the nutrients leach out and go to waste.

Actually, carbon can be delivered through the root membrane but it requires a carrier. A phospholipid might be able to carry carbon across, I think. It's like chelates for iron and other nutrients plants have trouble absorbing. The chelate grabs the nutrient, then the plant grabs the chelate, which is much easier to get ahold of. The chelate sort of "sneaks" the nutrient across and then goes back out into the rooting media to find more of whatever it helps move.

TBH I'm not quite this smart - I asked AN's Erik Biksa about this and that's what he told me.

Interesting! I'm actually an old fart who was taught the old kinds of truth.
Well, never too old to learn.

My grandpa liked to say that the day you stop learnin' is the day they plant you six feet down.

But unless I'm mistaken most of the carbon in plants comes from CO2. It's possible to absorb it through the roots, but they do it mainly through the leaves with photosynthesis.