Superstate Quantum Mechanics

We introduce Superstate Quantum Mechanics (SQM) as a theory that considers states in Hilbert space subject to multiple quadratic constraints. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization. In its simplest form, SQM considers states in the form of unitary operators, where the quadratic constraints are conditions of unitarity. In this case, the stationary SQM problem is a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. The SQM stationary problem is equivalent to a new algebraic problem that we address in this paper. The SQM non-stationary problem considers the evolution of a quantum system itself, distinct from the explicit time dependence of the Hamiltonian, $H(t)$. Two options for the SQM dynamic equation are considered: (1) within the framework of linear maps from higher-order quantum theory, where 2D-type quantum circuits are introduced to transform one quantum system into another; and (2) in the form of a Gross-Pitaevskii-type nonlinear map. Although no known physical process currently describes such 2D dynamics, this approach naturally bridges direct and inverse quantum mechanics problems, allowing for the development of a new type of computer algorithms. Beyond computer modeling, the developed theory could be directly applied if or when a physical process capable of solving a quantum inverse problem in a single measurement act (analogous to how an eigenvalue arises from a measurement in traditional quantum mechanics) is discovered in the future.