  Recent experimental studies have shown that the relaxation time ($T_1$) and
the dephasing time ($T_2$) of superconducting qubits fluctuate considerably
over time. To appropriately consider this time-varying nature of the $T_1$ and
$T_2$ parameters, a new class of quantum channels, known as Time-Varying
Quantum Channels (TVQCs), has been proposed. In previous works, realizations of
multi-qubit TVQCs have been assumed to be equal for all the qubits of an error
correction block, implying that the random variables that describe the
fluctuations of $T_1$ and $T_2$ are block-to-block uncorrelated, but qubit-wise
perfectly correlated for the same block. Physically, the fluctuations of these
decoherence parameters are explained by the incoherent coupling of the qubits
with unstable near-resonant two-level-systems (TLS), which indicates that such
variations may be local to each of the qubits of the system. In this article,
we perform a correlation analysis of the fluctuations of the relaxation times
of multi-qubit quantum processors ibmq\_quito, ibmq\_belem, ibmq\_lima,
ibmq\_santiago and ibmq\_bogota. Our results show that it is reasonable to
assume that the fluctuations of the relaxation and dephasing times of
superconducting qubits are local to each of the qubits of the system. Based on
these results, we discuss the multi-qubit TVQCs when the fluctuations of the
decoherence parameters for an error correction block are qubit-wise
uncorrelated (as well as from block-to-block), a scenario we have named the
Fast Time-Varying Quantum Channel (FTVQC). Furthermore, we lower bound the
quantum capacity of general FTVQCs based on a quantity we refer to as the
ergodic quantum capacity. Finally, we use numerical simulations to study the
performance of quantum error correction codes (QECC) when they operate over
FTVQCs.
