



01748nas a2200133   4500008004100000245008100041210006900122260001400191520131400205100001601519700002701535700001501562856003701577       2022                        eng d00aA Convergence Theory for Over-parameterized Variational Quantum Eigensolvers0 aConvergence Theory for Overparameterized Variational Quantum Eig  c5/25/20223 a<p>The Variational Quantum Eigensolver (VQE) is a promising candidate for quantum applications on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. Despite a lot of empirical studies and recent progress in theoretical understanding of VQE&#39;s optimization landscape, the convergence for optimizing VQE is far less understood. We provide the first rigorous analysis of the convergence of VQEs in the over-parameterization regime. By connecting the training dynamics with the Riemannian Gradient Flow on the unit-sphere, we establish a threshold on the sufficient number of parameters for efficient convergence, which depends polynomially on the system dimension and the spectral ratio, a property of the problem Hamiltonian, and could be resilient to gradient noise to some extent. We further illustrate that this overparameterization threshold could be vastly reduced for specific VQE instances by establishing an ansatz-dependent threshold paralleling our main result. We showcase that our ansatz-dependent threshold could serve as a proxy of the trainability of different VQE ansatzes without performing empirical experiments, which hence leads to a principled way of evaluating ansatz design. Finally, we conclude with a comprehensive empirical study that supports our theoretical findings</p>1 aYou, Xuchen1 aChakrabarti, Shouvanik1 aWu, Xiaodi  uhttps://arxiv.org/abs/2205.12481