Background🪶

Quantum Symplectic Structure

Though our system is comprised of discrete qubits (\(\therefore \, \dim{H} = 2^n\)), we leverage naturally emerging symplectic geometry that is admitted by quantum systems via a real-partition parameterization:

\[ \psi = u + i v \, \rightarrow \, Z = (u, v)^T \in \mathcal{R}^{2^{n+1}} \, , \]

and the fact that Schrödinger flow satisfies

\[ J \dot{Z} = \nabla_Z H(Z) \]

with canonical symplectic form

\[ J = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \, . \]

The steps we use to exploit these features are:

Computing time derivatives \(\dot{Z}\) during forward pass.
Adding a residual penalty

\[ \| \dot{Z} - \nabla_Z H(Z) \|^2 \]

to \(\mathcal{L}_\text{phys}\).
Interpreting POD modes in \(Z\)-space as symplectically-constrained trajectory variations.

This maintains low-fidelity character (e.g., no full geometric integrator) while grounding regularization in actual quantum mechanical geometry rather than heuristic smoothing.