What makes encryption "perfect"? This is not obvious, so I'll just skip ahead to the conclusion: encryption is perfect when it's impossible to distinguish which candidate plaintext led to a specific ciphertext without knowing the entire key.
It's easy to see that a one-time pad qualifies here. Even knowing all but one character of the plaintext doesn't give you any cryptographic way to determine what the last character is. The list of possible keys associating the known plaintext with the current ciphertext allows for every single character in the unknown position. This isn't a matter of computational power, it's an information-theoretic limit. You just don't have enough information (from the cryptography, anyway) to prefer any choice over another.
You can apply a pigeonhole principle argument here and see that this requires the key to have the same size as the message. (Something along the lines of each key specifying a permutation between messages of the same length, and you need the key size to be big enough to specify every possible permutation for that message length.)
That argument can then be used to demonstrate the information-theory quality of the encryption is related to the ratio between the lengths of the message and the key.
When you look at modern cryptography, you can see that we're not depending on information-theoretic security at all. If you had all but one characters of the plaintext and all the ciphertext, it's quite likely there'd only be one key that matches the known permutation. We've gone in a different direction: computational security. It should require an implausible amount of computation to derive what the key is, even when you know matching plaintext and ciphertext.
It's certainly more convenient than moving terabytes of key around, and computational limits are actually pretty compelling, as long as flaws aren't found in the construction.
But it lacks the satisfaction of provably unbreakable.