General Entropic Constraints on Calderbank-Shor-Steane Codes within Magic Distillation Protocols

Magic states are fundamental building blocks on the road to fault-tolerant quantum computing. Calderbank-Shor-Steane (CSS) codes play a crucial role in the construction of magic distillation protocols. Previous work has cast quantum computing with magic states for odd dimension d within a phase-space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions. Here we extend this framework to the important d=2 qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of outperforming previous monotone bounds in regimes of practical interest. Moreover, in the case of CSS-code projections, we arrive at a novel cutoff result on the code length n of the CSS code in terms of parameters characterizing a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs to consider only CSS codes below a threshold number of qubits. These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.