An application of Khovanov homology to quantum codes

TL;DR

This paper introduces new families of LDPC quantum error-correcting codes derived from Khovanov homology, demonstrating their asymptotic parameters and potential for quantum information protection.

Contribution

It applies Khovanov homology to construct novel LDPC quantum codes with specific asymptotic properties, expanding the toolkit for quantum error correction.

Findings

Unknot codes with asymptotic parameters [[3^(2l+1)/sqrt(8πl);1;2^l]]

Unlink codes with asymptotic parameters [[sqrt(2/2πl)6^l;2^l;2^l]]

(2,l)-torus link codes with d_n>√n/1.62

Abstract

We use Khovanov homology to define families of LDPC quantum error-correcting codes: unknot codes with asymptotical parameters [[3^(2l+1)/sqrt(8{\pi}l);1;2^l]]; unlink codes with asymptotical parameters [[sqrt(2/2{\pi}l)6^l;2^l;2^l]] and (2,l)-torus link codes with asymptotical parameters [[n;1;d_n]] where d_n>\sqrt(n)/1.62.