Redefining the Mathematical Foundations of Quantum Computing to Significantly Expand the Capability of Quantum Computers

Quantum computing (defined on the mathematical framework of complex numbers) is limited by its inability to divide by zero. This curtails the ability of quantum computers to handle singularities and infinities in simulations and solve ill-posed problems. This paper aims to define three new but distinct mathematical frameworks for building quantum computing architectures to expand the capabilities of quantum computers by enabling division by zero in quantum calculations. To do this, semi-structured complex numbers were partitioned into 3 distinct subsets of numbers: i-numbers, p-numbers and k-numbers. For each subset a novel set of Pauli matrices (representing quantum gates) were derived. The matrices and subsets were used to construct four mathematical Frameworks (111-Framework, 211-Framework, 121-Framework and 112-Framework). To demonstrate the utility of the frameworks the 211-Framework was selected, a universal gate set (from which all other quantum gates and circuits can be built) was defined, a novel Bloch sphere was constructed and the solution to an ill-defined problem was demonstrated. This paper provides a rigorous, and consistent way of incorporating division by zero into quantum computing and extends the computational reach of quantum computers.