Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies

Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on the manifolds of unitary and isometric matrices as well as the cone of positive-definite matrices. Combining this approach with the up-to-date computational methods of automatic differentiation, we demonstrate the efficacy of the Riemannian optimization in the study of the low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of a tensor network in the form of the multiscale entanglement-renormalization ansatz, preparation of arbitrary states (including highly entangled ones) in the circuit implementation of quantum computation, decomposition of quantum gates, and tomography of quantum states. Universality of the developed approach together with the provided open source software enable one to apply the Riemannian optimization to complex quantum architectures well beyond the listed problems, for instance, to the optimal control of noisy quantum systems.

Automated summary

The use of Riemannian optimization in quantum physics and quantum information science has shown effectiveness in solving optimization problems with constraints, such as low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of tensor networks, preparation of arbitrary quantum states, decomposition of quantum gates, and tomography of quantum states. The universality of this approach allows for its application to complex quantum architectures beyond the problems listed, including optimal control of noisy quantum systems.