General entropic constraints on CSS codes within magic distillation protocols

Abstract

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.

arXiv comment34 pages, 7 figures. Comments welcome! v2 clarifies the definition of CSS circuits and completes the proof that all CSS magic distillation protocols can be decomposed in terms of CSS code projections; v3 clarifies that our upper bounds are constraints on any code projection protocol able to carry out a desired distillation process, and are not proof that such a protocol exists

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