General entropic constraints on CSS codes within magic distillation protocols

TL;DR

This paper extends a phase space framework to the qubit case, deriving entropic bounds on CSS code-based magic state distillation that outperform previous bounds and impose practical code length limits.

Contribution

It introduces a novel phase space approach for qubit CSS codes, providing new distillation bounds and a cut-off on code length based on entropic constraints.

Findings

Outperforms previous monotone bounds in practical regimes

Establishes a code length threshold for effective distillation

Provides entropic constraints based on stochastic representations

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.