Set of steps that cannot be implemented on a Turing machine
The Church&ndash;Turing thesis states that every algorithm can be implemented on a Turing machine. It has stood the test of time. In particular, I believe that the answer to your question is negative.

The appearance of quantum algorithms, in particular Shor's factorization algorithm, has called into question the efficient Church&ndash;Turing thesis (sometimes known as the Cook&ndash;Karp thesis), which states that $\mathsf{P}$ captures the class of efficiently solvable problems; but quantum computers are known to compute the same class of languages as classical computers.