  Magic states are fundamental building blocks on the road to fault-tolerant
quantum computing. 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 out-performing previous monotone bounds in
regimes of practical interest. Moreover, in the case of CSS code projections,
we arrive at a novel cut-off result on the code length $n$ of the CSS code in
terms of parameters characterising a desired distillation, which implies that
for fixed target error rate and acceptance probability, one needs only consider
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.

Quantum Physics