EGT (Exact Geodesic Transport)¶

Implements EGT (Exact Geodesic Transport), a Riemannian optimizer for variational quantum algorithms that steps along exact geodesics on the quantum state manifold.

EGT works on the hypersphere \(\mathbb{S}^{d-1}\) of \(\ell_2\)-normalized state amplitudes \(x_t = \{x_j(\theta_t)\}\) rather than directly in parameter space. At each step it forms the natural-gradient velocity in the tangent space, then transports the state along the great-circle geodesic generated by that velocity via the closed-form exponential map. Because the manifold is a sphere, the geodesic is exact (no retraction approximation), and the new circuit parameters are recovered from the transported state. A conjugate-gradient variant (EGT-CG) accumulates a transported search direction with a coefficient \(\beta_t\) fixed by the strong Wolfe conditions.

\[ \begin{aligned} v_t &= -J(\theta_t)\, g^{-1}(\theta_t)\, (\partial_\theta \mathcal{L})_{\theta_t},\\ x_{t+1} &= \mathrm{ExpMap}_{x_t}(\eta_t v_t) = \cos(\eta_t \lVert v_t\rVert)\, x_t + \sin(\eta_t \lVert v_t\rVert)\, \frac{v_t}{\lVert v_t\rVert},\\ \theta_{t+1} &= \theta(x_{t+1}). \end{aligned} \]

where \(\theta\) are the variational parameters, \(x_t\) the normalized quantum state, \(\mathcal{L}(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle\) the energy loss, \(g^{-1}\) the inverse Riemannian (quantum geometric tensor) metric, \(J\) the Jacobian of the coordinate map, \(v_t\) the natural-gradient tangent velocity, and \(\eta_t\) the learning rate.

Reference: André J. Ferreira-Martins, Renato M. S. Farias, Giancarlo Camilo, Thiago O. Maciel, Allan Tosta, Ruge Lin, Abdulla Alhajri, Tobias Haug, Leandro Aolita, "Variational quantum algorithms with exact geodesic transport", arXiv 2025. https://arxiv.org/abs/2506.17395

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