# Superstate Quantum Mechanics

arXiv: https://arxiv.org/abs/2502.00037 · HTML: https://mlsanity.com/paper/2502.00037/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 70 references that are in the map.
- First posted on arXiv 25 Jan 2025.

## Builds on (most cited first, 11 of 11)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Learning Unitary Operators with Help From u(n)](https://mlsanity.com/paper/1607.04903/) (2016)
- [Optimal quantum dataset for learning a unitary transformation](https://mlsanity.com/paper/2203.00546/) (2022)
- [Gradient-descent quantum process tomography by learning Kraus operators](https://mlsanity.com/paper/2208.00812/) (2022)
- [On The Radon-Nikodym Spectral Approach With Optimal Clustering](https://mlsanity.com/paper/1906.00460/) (2019)
- [Partially Unitary Learning](https://mlsanity.com/paper/2405.10263/) (2024)
- [Quantum Channel Learning](https://mlsanity.com/paper/2407.04406/) (2024)
- [Introducing the quadratically-constrained quadratic programming framework in HPIPM](https://mlsanity.com/paper/2112.11872/) (2021)

## Built on by (most cited first, 1 of 1)

- [Semidefinite Programming for Quantum Channel Learning](https://mlsanity.com/paper/2601.12502/) (2026)

## Abstract (arXiv)

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.