  Recently, a class of fractal surface codes (FSCs), has been constructed on
fractal lattices with Hausdorff dimension $2+\epsilon$, which admits a
fault-tolerant non-Clifford CCZ gate. We investigate the performance of such
FSCs as fault-tolerant quantum memories. We prove that there exist decoding
strategies with non-zero thresholds for bit-flip and phase-flip errors in the
FSCs with Hausdorff dimension $2+\epsilon$. For the bit-flip errors, we adapt
the sweep decoder, developed for string-like syndromes in the regular 3D
surface code, to the FSCs by designing suitable modifications on the boundaries
of the holes in the fractal lattice. Our adaptation of the sweep decoder for
the FSCs maintains its self-correcting and single-shot nature. For the
phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder
for the point-like syndromes. We report a sustainable fault-tolerant threshold
($\sim 1.7\%$) under phenomenological noise for the sweep decoder and the code
capacity threshold (lower bounded by $2.95\%$) for the MWPM decoder for a
particular FSC with Hausdorff dimension $D_H\approx2.966$. The latter can be
mapped to a lower bound of the critical point of a confinement-Higgs transition
on the fractal lattice, which is tunable via the Hausdorff dimension.
