Quantum information processing is a critical part of the development of future computers, quantum computers, and quantum algorithms, where elementary particles such as photons and electrons can be applied in optomagnetic or optoelectronic devices. The computational physics behind these emerging approaches is also experiencing dramatic developments. In this paper I report on the most recent mathematical basics for quantum algorithms and quantum computing approaches. Some of these described approaches show intriguing methods for determining the states and wavefunction properties for anyons, bosons, and fermions in quantum wells that have been developed in the last years. The study also shows approaches based on N-quantum states and the reduced 1- and 2-fermion picture, which can be used for developing models for anyons and multi-fermionic states in quantum algorithms. Also, antisymmetry and generalized Pauli constraints have been given particular emphasis and include establishing a basis for pinning and quasipinning for exploring the symmetric states of many-fermionic subsets, as a foundation for quantum information processing, and are here briefly revisited. This study summarizes the developments in recent years of an advanced and important field for future computational techniques.