  The sensitivity of quantum systems to external disturbances is a fundamental
problem for the implementation of functional quantum devices, quantum
information and computation. Based on remarkable experimental progress in
optics and ultra-cold gases, we study the consequences of a short-time
(instantaneous) noise while an intensity-dependent phase acquisition is
associated with a qubit propagating on $N$-cycle. By employing quantum
coherence measures, we report emerging unstable regimes in which hitherto
unknown quantum walks arise, such as self-focusing and breathing dynamics. Our
results unveil appropriate settings which favor the stable regime, with the
asymptotic distribution surviving for weak nonlinearities and disappearing in
the thermodynamic limit with $1/N$. The diagram showing the threshold between
different regimes reveals the quantum gates close to Pauli-Z as more
noise-tolerant. As we move towards the Pauli-X quantum gate, such aptness
dramatically decreases and the threshold to self-focusing regime becomes almost
unavoidable. Quantum gates close to Hadamard exhibit an unusual aspect, in
which an increment of the nonlinear strength can remove the dynamics from
self-focusing regime.
