paper

General entropic constraints on CSS codes within magic distillation protocols

arXiv:2211.07535 · doi:10.1103/PRXQuantum.4.020359

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 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 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 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.

34 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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