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### Necessity for discarding errored magic state distillation

The reason you discard them is because this makes distillation much more efficient. For example, consider 15-to-1 T state distillation using the Reed-Solomon code. This code has distance 3, so you can ...
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Let's assume that we do the move without any errors. We denote the state before the move as $|\psi_L\rangle$, so that $$Z_L|\psi_L\rangle=|\psi'_L\rangle \\ X_L|\psi_L\rangle=|\psi''_L\rangle \\$$ ...