Conformal or near conformal Quantum Field Theories QFT) would benefit from a rigorous non-perturbative lattice formulation beyond the flat Euclidean space, $\mathbb R^d$. Although all UV complete QFT are generally acknowledged to be perturbatively renormalizable on smooth Riemann manifolds, non-perturbative realization on simplicial lattices (triangulation) encounter difficulties as the UV cut-off is removed. We review the Quantum Finite Element (QFE) method that combines classical Finite Element with new quantum counter terms designed to address this.
The construction for maximally symmetric spaces ($\mathbb S^d$, $\mathbb R\times \mathbb S^{d-1}$ and $\mathbb Ad\mathbb S^{d+1}$) is outlined with numerical tests on $\mathbb R\times\mathbb S^2$ and a description
of theoretical and algorithmic challenges for d = 3, 4 QFTs.