  Quantum errors in noisy environments remain a major obstacle to advancing
quantum information technology. In this work, we expand a recently developed
geometric framework, originally utilized for analyzing noise accumulation and
creating dynamical error-correcting gates at the control pulse level, to now
study noise dynamics at the quantum circuit level. Through a geometric
perspective, we demonstrate how circuit noise robustness can be enhanced using
twirling techniques. Additionally, we show that circuits modified by random
twirling correspond to random walk trajectories in this geometric framework,
and provide a fresh perspective on randomized compiling by analytically
deriving the perturbative expression for the resultant Pauli noise channel. We
also illustrate that combining robustness optimization strategies at both the
control pulse and circuit levels can significantly boost overall circuit
fidelity even further through numerical examples. This research illuminates
pathways to achieving noise-resistant quantum control beyond mere optimization
of control pulses.
