Permutation-invariant quantum codes

TL;DR

This paper constructs permutation-invariant quantum codes using combinatorics and operator theory, capable of correcting multiple errors in quantum systems modeled by the Heisenberg ferromagnet, with potential applications in quantum error correction.

Contribution

It introduces new families of permutation-invariant quantum codes with quadratic length that can perfectly or approximately correct multiple errors, extending existing quantum error correction criteria.

Findings

Codes of length proportional to t^2 correct t arbitrary errors

Codes approximately correct t spontaneous decay errors

Performance analysis extends Knill-Laflamme criteria

Abstract

A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground-space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to $t^2$; one family of codes perfectly corrects arbitrary weight $t$ errors, while the other family of codes approximately correct $t$ spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.