Scalable quantum simulation by reductions and decompositions through the Id-operator

One of the main obstacles for the adoption of quantum algorithm simulation is the exponential increase in temporal and spatial complexities, due to the expansion of transformations and read/write memory states by using tensor product in multi-dimension applications. Reduction and decomposition optimizations via the Id-operator provide a smart and appropriate storage and distribution of quantum information. Reductions are achieved by avoiding replication and sparsity inherited from such operators. By using decompositions, applications may be divided into sub-steps to store only distinct values from Id-operators, instead of executing quantum transformations in a single step. Additional optimizations based on mixed partial processes provide control over increase in read/write memory states in quantum transformations, also contributing to increase the scalability regarding hardware-GPUs memory limit. Hadamard and Discret Quantum Fourier Transforms were simulated up to 28 qubits applications over a single GPU with drastic temporal complexity reduction and simulation time.