Local distillation from Reed Muller codes unfolding

TL;DR

This paper extends the algebraic understanding of Reed Muller code-based distillation factories, presenting local layouts for stabilizers and demonstrating significant fidelity improvements in output states.

Contribution

It introduces a generalized algebraic framework for Reed Muller distillation factories and designs local layouts for different distances, improving state fidelity.

Findings

2D layout achieves output infidelity of 8.256e-9 from input p=1e-3

3D layout achieves output infidelity of 1.1811e-17 from input p=1e-3

Generalization enhances the understanding of Reed Muller code distillation processes.

Abstract

We generalize the unfolding of a Reed Muller distillation factory of Ruiz et. al. by exhibiting the algebraic structure that the unfolding is based on. We describe a 2D local layout for the Z stabilizers of a distance 4 Reed Muller distillation factory and a 3D local layout for the Z stabilizer of a distance 4 and a distance 7 Reed Muller distillation factory. Given input T states with infidelities $p=10^{-3}$, the 2D local distillation factory with distance 4 outputs a CCZ state with infidelity $p=8.256 \times 10^{-9}$ and the 3D local distillation factory with distance 7 outputs a T state with infidelity $p=1.1811 \times 10^{-17}$.