Magic state distillation with punctured polar codes

TL;DR

This paper introduces a novel scheme for magic state distillation using punctured polar codes, leveraging algebraic descriptions and efficient decoding to achieve high-performance quantum error correction.

Contribution

It constructs tri-orthogonal quantum codes based on polar codes with efficient decoding, enabling improved magic state distillation for quantum computing.

Findings

Codes scale as O(N^{0.8}) for erasure channel

Bit error rates drop to 8×10^{-28} for erasure channel

Bit error rates drop to 7×10^{-15} for dephasing channel

Abstract

We present a scheme for magic state distillation using punctured polar codes. Our results build on some recent work by Bardet et al. (ISIT, 2016) who discovered that polar codes can be described algebraically as decreasing monomial codes. Using this powerful framework, we construct tri-orthogonal quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic states for the $T$ gate. An advantage of these codes is that they permit the use of the successive cancellation decoder whose time complexity scales as $O(N\log(N))$. We supplement this with numerical simulations for the erasure channel and dephasing channel. We obtain estimates for the dimensions and error rates for the resulting codes for block sizes up to $2^{20}$ for the erasure channel and $2^{16}$ for the dephasing channel. The dimension of the triply-even codes we obtain is shown to scale like $O(N^{0.8})$ for the binary erasure channel at noise rate $0.01$ and $O(N^{0.84})$ for the dephasing channel at noise rate $0.001$. The corresponding bit error rates drop to roughly $8\times10^{-28}$ for the erasure channel and $7 \times 10^{-15}$ for the dephasing channel respectively.