Natural parametrized quantum circuit

Noisy intermediate-scale quantum computers are useful for various tasks such as state preparation and variational algorithms. However, the non-Euclidean geometry of parameterized circuits is detrimental these applications. Here we introduce natural circuit (NPQC) that can be initialized with a Euclidean geometry. The initial training algorithms substantially sped up gradient equivalent to gradient. Further, show how estimate parameters NPQC by sampling circuit, which could used benchmarking or calibrating NISQ hardware. For general class circuits, has minimal Cram\'er-Rao bound, highlights its potential metrology. Finally, generate arbitrary superpositions two states NPQCs tasks. Our results enhance currently available processors.