  Self-correcting quantum memories demonstrate robust properties that can be
exploited to improve active quantum error-correction protocols. Here we propose
a cellular automaton decoder for a variation of the color code where the bases
of the physical qubits are locally rotated, which we call the XYZ color code.
The local transformation means our decoder demonstrates key properties of a
two-dimensional fractal code if the noise acting on the system is infinitely
biased towards dephasing, namely, no string-like logical operators. As such, in
the high-bias limit, our local decoder reproduces the behavior of a partially
self-correcting memory. At low error rates, our simulations show that the
memory time diverges polynomially with system size without intervention from a
global decoder, up to some critical system size that grows as the error rate is
lowered. Furthermore, although we find that we cannot reproduce partially
self-correcting behavior at finite bias, our numerics demonstrate improved
memory times at realistic noise biases. Our results therefore motivate the
design of tailored cellular automaton decoders that help to reduce the
bandwidth demands of global decoding for realistic noise models.
