Let's define a "card deck" as a sequence of the first $x$ natural numbers, and a "swap" as the creation of a new sequence where the $i$th and $j$th member of the original sequence are exchanged e.g. swapping cards $2$ and $3$ so that $[1, 2, 3, 4 ] \longmapsto [1, 3, 2, 4]$.
My question is, how many swaps (where i and j are randomly chosen) must be made before a sequence is created that can be pronounced random? Is there a faster way to achieve randomness if i and j and not chosen randomly?
EDIT: I suppose I don't care how random it is, though let's say the sequence is a real deck of cards; the randomness should be close to what you would get with $7$ good riffles (see Shuffling). Even better would be a way to control the degree of randomness, i.e. estimate $z$, where your odds of predicting the location of a particular card in the deck are $\frac1x + z$.
Note: It seems that picking i and j randomly is not the best way to achieve randomness. See Fisher-Yates shuffle.