Qudit stabilizers beyond the free case and the twisted Kitaev model

TL;DR

This paper extends the stabiliser formalism for qudits of arbitrary dimension, generalising known results to non-free cases and applying the framework to the qudit Kitaev model and its variants.

Contribution

It generalises the stabiliser formalism beyond the free case and describes the structure of protected spaces and quotient groups for qudits of arbitrary dimension.

Findings

The stabiliser formalism remains valid for free qudit stabilisers without non-trivial scalars.

The structure of protected spaces is characterized as tensor products of smaller qudit spaces.

The approach applies to the qudit Kitaev model and its twisted variants.

Abstract

We study the stabiliser formalism for qudits of arbitrary dimension $d$. In the free case, we show that the basic theorem of the stabiliser formalism remains valid: if the stabiliser subgroup $H$ is free as a $Z/dZ$-module and contains no non-trivial scalars, then the protected space $V^H$ is naturally identified with the state space of a smaller number of qudits of the same dimension, and the quotient $N(H)/H$ is identified with the Pauli group on a smaller number of qudits. We then remove the freeness assumption and describe the resulting structure in general. In this case, the protected space is identified with a tensor product of qudit spaces of possibly smaller dimensions, and the quotient $N(H)/H$ is described by a corresponding product of qudit Pauli groups, possibly of smaller dimensions, over a common center. We also characterise the shifted free case, which is exactly the situation in which $N(H)/H$ is again an ordinary qudit Pauli group. Our approach is algebraic and uniform, and applies in particular to the qudit Kitaev model and to its shifted and twisted variants.