  Progress in fabrication of semiconductor and superconductor qubits has
greatly diminished the number of decohering defects, thus decreasing the
devastating low-frequency $1/f$ noise and extending the qubits' coherence times
(dephasing time $T_2^*$ and the echo decay time $T_2$). However, large
qubit-to-qubit variation of the coherence properties remains a problem, making
it difficult to produce a large-scale register where all qubits have a
uniformly high quality. In this work we show that large variability is a
characteristic feature of a qubit dephased by a sparse bath made of many ($n\gg
1$) decohering defects, coupled to the qubit with similar strength. We model
the defects as two-level fluctuators (TLFs) whose transition rates $\gamma$ are
sampled from a log-uniform distribution over an interval
$[\gamma_{m},\gamma_M]$, which is a standard model for $1/f$ noise. We
investigate decoherence by such a bath in the limit of high-quality qubit,
i.e.\ when the TLF density $d$ is small (the limit of sparse bath, with
$d=n/w\ll 1$, where $n$ is the number of TLFs and
$w=\ln{[\gamma_M/\gamma_{m}]}$ is the log-width of the distribution). We show
that different realizations of the bath produce very similar noise power
spectra $S(f)\sim 1/f$, but lead to drastically different coherence times
$T_2^*$ and $T_2$. Thus, the spectral density $S(f)$ does not determine
coherence of a qubit coupled to a sparse TLF bath, as opposed to a dense bath;
instead, decoherence is controlled by only a few exceptional fluctuators,
determined by their value of $\gamma$. We show that removing only two of these
TLFs greatly increases $T_2$ and $T_2^*$ times. Our findings help theoretical
understanding and further improvements in the coherence properties of
semiconductor and superconductor qubits, battling the $1/f$ noise in these
platforms.
