Quantum Algorithms on Symplectic State Manifolds

We introduce QuanTY, an algorithmic framework for quantum computation grounded in Quantum Inner State Manifolds (QISMs) and Topologically Transient (TT) structures. In QuanTY, a Y-qubit quantum state is modeled as a $2Y$-dimensional symplectic manifold, enabling quantum evolution to be formulated as continuous Hamiltonian flow rather than discrete circuit dynamics. Quantum gates and algorithms are realized as structured symplectomorphisms, while topological and homological invariants-most notably Floer homology and quantum cohomology-are continuously tracked as computational witnesses throughout the evolution. A central feature of the framework is the introduction of TT structures: algorithmically controlled topological transitions designed to resolve geometric degeneracies, non-transversality, and numerical instability, while provably preserving all algorithmically relevant invariant data. This mechanism enables adaptive computation without measurement-induced collapse and provides intrinsic stability control absent in fixed-topology computational models. We establish formal correctness and stability results for QuanTY computations and derive polynomial complexity bounds for TT-controlled evolution on product QISMs of the form (S²)Y. Canonical quantum algorithms are reformulated within this framework, including a complete geometric realization of Grover's search algorithm, demonstrating that standard quantum speedups are fully recovered within the symplectic-topological setting. Numerical simulations and a reference implementation validate the theoretical analysis and demonstrate controlled scaling for systems of up to $Y = 16$ qubits. Taken together, these results position QuanTY as a unified geometric-topological paradigm for quantum computation, bridging symplectic geometry, homological methods, and practical algorithmic realization, while offering a stable and adaptive alternative to conventional circuit-based quantum models.