Six-semion stabilizer code[1]

Description

A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) and two qudits per edge of the square lattice that realizes the six-semion topological order. The code is obtained from two copies of the \(\mathbb{Z}_4\) toric code by condensing the bosons \(e_1^2 m_1^2\) and \(e_1^2 e_2^2 m_2^2\) [1,2]. Its stabilizer generators are four terms from the stabilizer group of the two toric codes together with four condensation terms [2; Sec. VI.C].

The six-semion anyon theory is the Abelian TQD of \(G=\mathbb{Z}_2\times\mathbb{Z}_2\) with both Type-I cocycles and the Type-II cocycle [1]. Its 16 anyons obey \(\mathbb{Z}_4\times\mathbb{Z}_4\) fusion rules and consist of four bosons, six semions, and six anti-semions [1]. Two basis anyons of order four have topological spin \(-i\) and mutual braiding phase \(i\) [2; Table V].