Fractal Properties of Magic State Distillation

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.