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.

翻译：本文提出超态量子力学（SQM），该理论将希尔伯特空间中的态视为受多重二次约束的对象，且“能量”同样表示为这些态的二次函数。传统量子力学对应于波函数归一化的单一二次约束，其能量表示为包含哈密顿量的二次型。当SQM将态表示为酉算子时，其定态问题转化为量子逆问题，在物理学、机器学习和人工智能领域具有多重应用。任何SQM定态问题均等价于本文探讨的新代数问题。非定态SQM问题考虑系统自身的演化，涉及与定态情形相同的“能量”算子。我们探讨了SQM动力学方程的两种可能形式：（1）基于高阶量子理论中的线性映射框架，其中二维型量子电路实现量子系统间的相互转换；（2）采用Gross-Pitaevskii型非线性映射形式。尽管目前尚未发现描述此类二维动力学的物理过程，但该方法天然桥接了正向量子力学与逆问题，为发展新型计算机算法提供了可能。作为该理论立即可行的实际应用，我们探讨了将量子通道作为经典计算模型的方案，此类计算可在经典计算机上实现。