The Supercomputer, also referred as the "Supercalculator",is a quantum supercomputer designed to hold Lyoko, X.A.N.A., and countless actual individuals in its data at the same time. It is housed in theSupercomputer Roombelow the Laboratory.

In Code Lyoko Evolution, the final episode (Ultimate Mission), the supercomputer was shut down again.

The Supercomputer was built by Franz Hopper for the purpose of destroying Carthage. During construction of the supercomputer, Franz Hopper used the Time reversion feature 2,564 times to complete work on the computer. Each time reversion doubles the processing power of the supercomputer, and by extension, X.A.N.A., the Supercomputer is powered by an uranium rod.

The actual Supercomputer is at the lowest level of the Factory where it rises from a hatch with a kind of liquid, most likely liquid nitrogen used for cooling, in the center of the room. The Supercomputer also has a monitoring port a few stories above, but only Jeremie and Aelita can fully understand its potential.

InThe Girl of the Dreams, Jeremy shoutsReturn to the past now! and aTime reversion initiates, even though he has been dragged away from the computer. This may mean that the supercomputer can take voice commands.

In the second season, X.A.N.A. and Aelita were both dependent to the supercomputer's survival. Should it have failed, or ran out of energy, X.A.N.A. would have been destroyed along with Aelita.To that regard, Team Lyoko did not shut down the Supercomputer because of Aelita's dependency to the computer.

In the series finale, the group finally shut down the Supercomputer. Also, in this episode, its hatch is seen with the Eye of X.A.N.A.on it.

The idea of the supercomputer doubling in power with every Return to the Past is probably erroneous - if the supercomputer added a qubit with every return trip, then after Franz was done its data-processing power would be laughable - most likely it takes multiple RTTPs to generate another qubit and increase the processing power. If the processing power truly did double every time, then the number would exceed ten googolplex.

The subject of the fuel rod is also of debate. Uranium's half-life means the battery would not expire for hundreds of years, also Duncan and Jeremie would have been severely poisoned from being around both enriched Uranium and the depleted fuel rod - therefore it seems Lead-210, a Uranium isotope with a twenty-two year half-life, is the used fuel - it only gives off particle radiation, which cannot penetrate the perspex sheath of the fuel rods, and which could be converted to energy by the same process we see in the scanners when a body is destroyed and rebuilt during virtualization and materialization, respectively.

The outer casing of the Supercomputer's resting place while dormant.

More here:

Supercomputer - Code Lyoko Wiki