Non-additivity of decoherence rates in superconducting qubits

We show that the relaxation and decoherence rates $T_1^{-1}$ and $T_2^{-1}$ of a qubit coupled to several noise sources are in general not additive, i.e., that the total rates are not the sums of the rates due to each individual noise source. To demonstrate this, we calculate the relaxation and pure dephasing rates $T_1^{-1}$ and $T_\phi^{-1}$ of a superconducting (SC) flux qubit in the Born-Markov approximation in the presence of several circuit impedances $Z_i$ using network graph theory and determine their deviation from additivity (the mixing term). We find that there is no mixing term in $T_\phi^{-1}$ and that the mixing terms in $T_{1}^{-1}$ and $T_2^{-1}$ can be positive or negative, leading to reduced or enhanced relaxation and decoherence times $T_1$ and $T_2$. The mixing term due to the circuit inductance $L$ at the qubit transition frequency $\omega_{01}$ is generally of second order in $\omega_{01}L/Z_i$, but of third order if all impedances $Z_i$ are pure resistances. We calculate $T_{1,2}$ for an example of a SC flux qubit coupled to two impedances.