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