The thing is with a sub carrier, it is a HVU, or a High Value Unit. If the presence of one is suspected, you can expect that a lot of enemy assets will be deployed to attack it.
A carrier is meant to be a beacon of permanent power projection; it is meant to bust its way into an area, and sustain a presence there. A submarine aircraft carrier cannot do either; it cannot break its way into an area and sustain a presence. Such a design, may I state this clearly, IS NOT PRACTICAL, even with the advances of technology seen today.
Well, at least for me it seems that a carrier exists to provide means to find and blow up things in the air, land and sea. In the distant future (where sub carriers clearly belong) the finding part might be as well provided by long range land based assets and space based assets. The blowing things up on land part is even today produced more cost-effectively by long range bombers and cruise missiles. The carrier air group cannot also strike much inland. With advent of better communications and precision guidance traditional CAS missions can be performed by naval gunfire closer to the shoreline and by bombers further inland.
At sea it is debatable if long range bombers equipped with suitable missiles might be just as able to perform ASUW strike mission as well if not even better than carrier air group. For ASW a carrier does not offer any capabilities which land based long range aircraft and normal DDG/FFG/CG helos cannotoffer.
For power projection a sub carrier might be as effective as a surface carrier. UK reportedly prevented Argentine invasion of Falklands during 1970's just by announcing their subs were in the area.
Finally, carriers are hideously expensive. To deliver any payload a carrier battle group needs some 9000 men to be placed under possible enemy attack. For me at least it seems that most of the carrier battle group is tied up to defend itself. To sustain one carrier battle group on position you need two other carrier groups as well, with a total of some 27000 men. To fly bombers halfway round the world with tanker support is surely expensive, but I would still guess it does not tie up as many men.
For me at least it seems that the only thing a traditional carrier is ultimately needed is establishment of air superiority in areas which are out of reach for land based fighters and friendly SAM cover. (Such as defence of Taiwan mission).
A sub carrier might be useful to launch air combat UCAV's fairly close to enemy territory. Furthermore, a sub carrier might be of some modular design in which aircraft hangars etc. assorted equipment might be also used for cruise missile cells if needed. As with UCAV's one does not need to care as much about safety issues these might be launched underwater in similar way to ballistic missiles. The problems start when UCAV's would land and replenish. For this the UCAV's might use satellite comms to call up an ELF transmitter to alert the sub. Even if the retrieval procedure would be very quick, the sub carrier would need a rudimentary air defense capabilities.
Obviously the sub might not hold neither as much planes, fuel and munitions as a traditional carrier in order to keep the sub carrier in size limits which would allow it to operate unescorted, like Soviet Project 941 class did. This might be countered by reducing the mission load (as proposed above) to air superiority, and by building more sub carriers as the crew (and thus money) savings would be significant.
Yes, this is quite futuristic but what does one do when sitting on a boring job shift...In reality, USN will be stuck with Nimitzes for next 50 years. China will probably build a boring traditional carrier if it does build one at all. EU will discuss the matter.