Is there a software algorithm that can generate a non-deterministic chaos pattern?
No deterministic algorithm can generate a non-deterministic output.

A non-deterministic algorithm (e.g., a NDFA) could, but they exist only in theory, not in practice.

An algorithm with access to a random number generator can generate a random output, and therefore its output is not deterministic.  For instance, in practice by reading from /dev/urandom or CryptGenRandom(), you can get random bits, which can be used to produce random output, or to seed a pseudorandom number generator.  (You don't need a quantum random number generator to get random numbers.)

Chaos theory is pretty much irrelevant here.