The Quantum Algorithm Zoo includes a host of algorithms for which Quantum Computing offers speedups (exponential, polynomial, etc). However, those speedups are based on asymptotic computational complexity (Big-O complexity).
For a realistic implementation on a quantum computer (or even a simulator), the algorithm might require other add-ons to get a satisfiable result. For e.g., for multiple quantum state tomography trials, or probabilistic cloning. I am not considering an ideal quantum computer, thus, overheads of physical to logical qubit mapping for Quantum Error Correction; overheads of nearest neighbour mapping; or experimental environmental errors are not considered. I am only considering the effect of projective measurement and no-cloning.
How can I compare a quantum algorithm taking into account such factors? The overheads might be polynomial, linear, or even constant, so asymptotically it will outperform the corresponding classical algorithm. But, asymptotically means, for a large enough problem size. I am interested in determining if that cross-over point (quantum supremacy problem size) is realistic for my case.
The specific algorithm I am currently working with is an improvement of Grover's search as proposed by Biham et. al, where the initial amplitude distribution is not uniform, and there are multiple solutions.