How are black-box oracles implemented in Hamiltonian simulation?

I am currently trying to decompose a hessian to a sum of unitaries $$H=\sum a_i U_i$$.

The papers VQLS and Black-box Hamiltonian Simulation state that it can be done, but requires the use of an oracle acting as

$$O_F \left| j, k \right> = \left| j, f(j, k)\right>$$ for any $$j \in \{1,... N\}$$ and $$k \in \{1, ..., D\}$$, where $$f(j,k)$$ gives the row index of the $$k$$th nonzero element of column $$j$$.

It is also stated in the paper that this is generally not hard to contruct, yet I don't have a clue how to do that.

So, is it that trivial to implement this oracle?

From my understanding, black boxes(oracle) which is the same idea as classical computer functions, is an idea of a group of unknown gates or things that are irreversible, that help transform a system from quantum state $$|x>$$ into $$|f(x)$$>, trough the evolution of quantum states, like Grover search make use of amplitude amplification, which is similar to an oracle, but try to amplitude average.For more detailed explanation of Grover search