# nLab qudit

Contents

### Context

#### Computation

intuitionistic mathematics

# Contents

## Idea

In quantum information theory and quantum computing, by a qdit (or qudit) one means a quantum state in a $d$-dimensional Hilbert space, for any natural number $d$.

Hence for fixed $d\in \mathbb{N}$, the quantum data type of qdits is the $d$-dimensional complex vector space equipped with a quantum measurement-basis

$\mathbb{C}^d \;\simeq\; \underset{n \in \{1, \cdots, d\}}{\bigoplus} \mathbb{C} \cdot \vert n \rangle \,.$
• For $d = 2$ one speaks of qbits, this is the original terminology;

• For $d = 3$ one also speaks of qtrits.

## Examples

### Anyon states

A key example of quantum gates which naturally act on qdits for $d \gt 2$ are anyon braid-gates in topological quantum computation:

For the potentially realistic case of Chern-Simons theory/WZW-model-controled anyons (such as su(2)-anyons), the elementary quantum logic gates act by the monodromy of the Knizhnik-Zamolodchikov connection on Hilbert spaces of conformal blocks, whose dimension $d$ is given by a Verlinde formula.

References which make this point explicit include Kolganov, Mironov & Morozov (2023).

General: