  Very little is known about the cost landscape for parametrized Quantum
Circuits (PQCs). Nevertheless, PQCs are employed in Quantum Neural Networks and
Variational Quantum Algorithms, which may allow for near-term quantum
advantage. Such applications require good optimizers to train PQCs. Recent
works have focused on quantum-aware optimizers specifically tailored for PQCs.
However, ignorance of the cost landscape could hinder progress towards such
optimizers. In this work, we analytically prove two results for PQCs: (1) We
find an exponentially large symmetry in PQCs, yielding an exponentially large
degeneracy of the minima in the cost landscape. Alternatively, this can be cast
as an exponential reduction in the volume of relevant hyperparameter space. (2)
We study the resilience of the symmetries under noise, and show that while it
is conserved under unital noise, non-unital channels can break these symmetries
and lift the degeneracy of minima, leading to multiple new local minima. Based
on these results, we introduce an optimization method called Symmetry-based
Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs. Our
numerical simulations show that SYMH improves the overall optimizer performance
in the presence of non-unital noise at a level comparable to current hardware.
Overall, this work derives large-scale circuit symmetries from local gate
transformations, and uses them to construct a noise-aware optimization method.
