  Magic state distillation protocols have a complicated non-linear nature.
Analysis of protocols is therefore usually restricted to one-parameter families
of states, which aids tractability. We show that if we lift this one-parameter
restriction and embrace the complexity, distillation exhibits fractal
properties. By studying these fractals we demonstrate that some protocols are
more effective when not restricted. Low fidelity states that are usually
worthless for distillation are now usable, and fewer iterations of the
protocols are needed to reach high fidelity.
