Superstate Quantum Mechanics

TL;DR

Superstate Quantum Mechanics (SQM) introduces a framework with multiple quadratic constraints on quantum states, enabling new inverse problems and computational methods that bridge traditional and nonlinear quantum dynamics.

Contribution

The paper develops a novel SQM framework with multiple quadratic constraints, addressing stationary and non-stationary problems, and proposes new algebraic and dynamic equations for quantum systems.

Findings

SQM generalizes traditional quantum mechanics with multiple constraints.

A new algebraic problem equivalent to stationary SQM is formulated.

Potential for quantum-inspired classical algorithms is demonstrated.

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.