Superstate Quantum Mechanics

TL;DR

SQM generalizes quantum mechanics by imposing multiple quadratic constraints on states, with energy defined as a quadratic functional; the stationary SQM problem becomes a quantum inverse problem tackled via a quadratic fidelity in the mapping operator \\mathcal{U}. The central development is the algebraic reformulation S\\mathcal{U} = \\lambda \\mathcal{U}, where the Hermitian matrix \\lambda and the operator \\mathcal{U} encode a hierarchy of stationary solutions, each with fidelity \\mathcal{F} = \\mathrm{Tr} \\lambda. The authors establish a canonical form for unitary channels when D = n, discuss density-of-states classifications, and propose both linear higher-order dynamics and nonlinear Gross–Pitaevskii–type dynamics for non-stationary SQM, including 2D circuit representations that transform quantum systems rather than states. The framework offers a pathway to classical simulations of quantum channels and potential ML applications by embedding inverse problems into a QCQP structure, with future work needed on multiple-solution convergence, non-stationary dynamics, and physical realizations of quantum inverse measurements.

Abstract

We introduce Superstate Quantum Mechanics (SQM), a theory that considers states in Hilbert space subject to multiple quadratic constraints, with ``energy'' also expressed as a quadratic function of these states. Traditional quantum mechanics corresponds to a single quadratic constraint of wavefunction normalization with energy expressed as a quadratic form involving the Hamiltonian. When SQM represents states as unitary operators, the stationary problem becomes a quantum inverse problem with multiple applications in physics, machine learning, and artificial intelligence. Any stationary SQM problem is equivalent to a new algebraic problem that we address in this paper. The non-stationary SQM problem considers the evolution of the system itself, involving the same ``energy'' operator as in the stationary case. Two possible 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 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. As an immediately available practical application of the theory, we consider using a quantum channel as a classical computational model; this type of computation can be performed on a classical computer.