I am reading the paper Restrictions on Transversal Encoded Quantum Gate Sets, Bryan Eastin, Emanuel Knill. I am unable to understand the following lines in the proof.
As the set of all unitary operators is a metric space, a finite number of unitary operators cannot approximate infinitely many to arbitrary precision
I need some references or hints as to understand how the above-quoted lines.
In their case, their finite set of unitary operators is closed under composition. They even have footnote that emphasizes this.
You can't approximate infinite set by some finite subset with the error that is less than half the minimum distance between elements in this finite subset. This is like trying to approximate every real number from $[0,1]$ by some finite subset with arbitrary precision. It's impossible.
Probably the easiest way to think about this is to consider an equivalent statement for the real numbers. Consider the range $[0,1]$, for instance. You're given a finite set of real numbers within that range. If you think about these values on the number line, it should be fairly obvious that there are necessarily points that are a finite distance away, so I cannot use members of this set as arbitrarily accurate approximations for any real number in the range. (In this case, if your set contains $n$ elements, it would be best to have them at values $(2i-1)/(2n)$ for $i=1$ to $n$ so that all real numbers in the range are within $1/(2n)$ of some element in the set. But if you try to bunch some closer together, obviously other have to get further apart).
You can make the same argument for any continuous space for which there is a distance measure.
