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Optimization with constraints is a typical problem in quantum physics andquantum information science that becomes especially challenging forhigh-dimensional systems and complex architectures like tensor networks. Herewe use ideas of Riemannian geometry to perform optimization on manifolds ofunitary and isometric matrices as well as the cone of positive-definitematrices. Combining this approach with the up-to-date computational methods ofautomatic differentiation, we demonstrate the efficacy of the Riemannianoptimization in the study of the low-energy spectrum and eigenstates ofmultipartite Hamiltonians, variational search of a tensor network in the formof the multiscale entanglement-renormalization ansatz, preparation of arbitrarystates (including highly entangled ones) in the circuit implementation ofquantum computation, decomposition of quantum gates, and tomography of quantumstates. Universality of the developed approach together with the provided opensource software enable one to apply the Riemannian optimization to complexquantum architectures well beyond the listed problems, for instance, to theoptimal control of noisy quantum systems.

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