Symplectic perspective to quantum computing for Hamiltonian systems

• URL: http://arxiv.org/abs/2604.10794v1
• Date: Sun, 12 Apr 2026 19:55:05 GMT
• Title: Symplectic perspective to quantum computing for Hamiltonian systems
• Abstract summary: This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems.<n>We establish an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold.<n>We show that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a two-fold way. The first part is devoted in establishing an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold. This correspondence enables a geometric quantization scheme that identifies a family of classical Hamiltonian systems admitting exponentially compressed quantum representations-appropriate for quantum simulation. In the second part we demonstrate that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding. This allows efficient quantum representation and parallel evolution of large phase-space ensembles, where entangled encodings provide exponential compression in ensemble size and enable quantum speed-ups in observable estimation via amplitude estimation techniques. For non-integrable systems, Lie canonical perturbation theory is incorporated to construct near-symplectic transformations that map dynamics to approximately integrable forms, preserving unitary evolution up to a controlled error. We derive the resulting quantum computational complexity of the proposed quantum-symplectic scheme, revealing both an exponential compression in memory requirements and a potential polynomial speed-up with respect to the system size. Finally, the transport evolution equation governing the quantum phase-space observables is obtained.

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