The article is specifically referring to symmetric encryption. It's not clear what algorithm is implied for 64 bits, as the AES standard only defined 128, 192, and 256 bit key sizes. An AES key is (when generated properly) a completely and truly random sequence of bits. To "crack" an AES key, one needs to have cipher text of a message or messages and attempt key possibilities (each possible key value, 2^128, 2^192, or 2^256 for the respective key size) which thru AES decryption yield plaintext which is correct (meaning the plaintext of the party who had encrypted the aforementioned cipher text).

For asymmetric keys, let's focus on RSA for example, the process of factoring (which is how one might "crack" an RSA public key into its private key components) has a computational hardness which is the basis of its security. Quantum computing poses the risk of being able to do this in a reasonable amount of time (whereas traditional computers can not).

NIST has been working with a number of organizations for the last couple of years to evaluate replacement algorithms which are believed to be "Quantum Resistant".