Connecting geometry and performance of two-qubit parameterized quantum circuits

Parameterized quantum circuits (PQCs) are a central component of many variational algorithms, yet there is lack understanding how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use Mannoury-Fubini-Study metric find simple equation relating Ricci scalar (geometry) and concurrence (entanglement). By calculating during eigensolver (VQE) optimization process, offers us new perspective why Quantum Natural Gradient outperforms standard gradient descent. argue that key Gradient's superior performance its ability regions high negative curvature early in process. These appear be important accelerating