  Correlated noise across multiple qubits poses a significant challenge for
achieving scalable and fault-tolerant quantum processors. Despite recent
experimental efforts to quantify this noise in various qubit architectures, a
comprehensive understanding of its role in qubit dynamics remains elusive.
Here, we present an analytical study of the dynamics of driven qubits under
spatially correlated noise, including both Markovian and non-Markovian noise.
Surprisingly, we find that, while correlated classical noise only leads to
correlated decoherence without increasing the quantum coherence in the system,
the correlated quantum noise can be exploited to generate entanglement. In
particular, we reveal that, in the quantum limit, pure dephasing noise induces
a coherent long-range two-qubit Ising interaction that correlates distant
qubits. In contrast, for purely transverse noise when qubits are subjected to
coherent drives, the correlated quantum noise induces both coherent symmetric
exchange and Dzyaloshinskii-Moriya interaction between the qubits, as well as
correlated relaxation, both of which give rise to significant entanglement.
Remarkably, in this case, we uncover that the system exhibits distinct
dynamical phases in different parameter regimes. Finally, we reveal the impact
of spatio-temporally correlated 1/f noise on the decoherence rate, and how its
temporal correlations restore lost entanglement. Our analysis not only offers
critical insights into designing effective error mitigation strategies to
reduce harmful effects of correlated noise, but also enables tailored protocols
to leverage and harness noise-induced correlations for quantum information
processing.
