If I understand you correctly, your goal is:
To choose some quantum algorithm (your question is: which algorithm would be good?)
Instead of running the quantum algorithm on a real quantum computer, you want to run a simulation of a quantum computer on a classical computer to simulate the execution of the quantum algorithm.
You want to optimize your classical simulator to make efficient use of the special capabilities offered by a High Performance Cluster (e.g. parallelism) and demonstrate that the simulation runs faster.
If so, it really does not matter which quantum algorithm you choose, as long as it involves many qubits and many gates.
You are probably best off choosing some very well-known algorithm that is well described in the literature, e.g. Grover or Shor.
Your simulation will involve multiplying very large matrices with each other.
The more qubits, the larger the matrices. The growth is exponential: if you have $N$ qubits, you will be multiplying $2^N$ by $2^N$ matrices.
The more gates, the more multiplications.
You will make life easier on yourself if you assume the qubits are perfect (no noise, no errors).
Since the main point of your exercise is to parallelize some code for an HPC, it probably doesn't matter that the simulation is less realistic.