Quantum natural gradient generalized to noisy and nonunitary circuits

Variational quantum algorithms are promising tools whose efficacy depends on their optimization method. For noise-free unitary circuits, the generalization of natural gradient descent has been introduced and shown to be equivalent imaginary time evolution: approach is effective due a metric tensor reconciling classical parameter space device's Hilbert space. Here we generalize consider arbitrary states (both mixed pure) via completely positive maps; thus our circuits can incorporate both imperfect gates fundamentally nonunitary operations such as measurements. We employ Fisher information (QFI) core in density operators. A modification error suppression by derangements (ESD) virtual distillation (VD) techniques enables an accurate experimentally efficient approximation QFI Hilbert-Schmidt using prior results dominant eigenvector noisy states. Our rigorous proof also establishes fundamental observation that geometry typical (approximately) identical either or characterized QFI. In numerical simulations demonstrate practicality confirm it significantly outperform other variational techniques.