  The qubit-mapping problem aims to assign and route qubits of a quantum
circuit onto a NISQ device in an optimized fashion, with respect to some cost
function. Finding an optimal solution to this problem is known to scale
exponentially in computational complexity; as such, it is imperative to
investigate scalable qubit-mapping solutions for NISQ computation. In this
work, a noise-aware heuristic qubit-assignment algorithm (which assigns initial
placements for qubits in a quantum algorithm to qubits on a NISQ device, but
does not route qubits during the quantum algorithm's execution) is presented
and compared against the optimal \textit{brute-force} solution, as well as a
trivial qubit assignment, with the aim to quantify the performance of our
heuristic qubit-assignment algorithm. We find that for small, connected-graph
algorithms, our heuristic-assignment algorithm faithfully lies in between the
effective upper and lower bounds given by the brute-force and trivial
qubit-assignment algorithms. Additionally, we find that the topological-graph
properties of quantum algorithms with over six qubits play an important role in
our heuristic qubit-assignment algorithm's performance on NISQ devices.
Finally, we investigate the scaling properties of our heuristic algorithm for
quantum processors with up to 100 qubits; here, the algorithm was found to be
scalable for quantum-algorithms which admit path-like graphs. Our findings show
that as the size of the quantum processor in our simulation grows, so do the
benefits from utilizing the heuristic qubit-assignment algorithm, under
particular constraints for our heuristic algorithm. This work thus
characterizes the performance of a heuristic qubit-assignment algorithm with
respect to the topological-graph and scaling properties of a quantum algorithm
which one may wish to run on a given NISQ device.
