A theory of quantum error correction for permutation-invariant codes

TL;DR

This paper develops a comprehensive theory for quantum error correction tailored to permutation-invariant codes, introducing algorithms based on symmetric group representation theory for efficient error correction.

Contribution

It provides the first general framework for error correction in permutation-invariant quantum codes, including algorithms using angular momentum measurements and quantum Schur transforms.

Findings

Algorithms can correct any correctible error on PI codes

Efficient correction algorithms for erasure and deletion errors

Use of representation theory and quantum transforms in error correction

Abstract

We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.