A Review of Galois Qudits

TL;DR

This review paper consolidates and formalizes the mathematical framework of Galois qudits, highlighting their structure, measurement, and applications in quantum error correction, especially over binary extension fields.

Contribution

It systematically collects and formalizes existing knowledge on Galois qudits, including definitions, Clifford hierarchies, stabilizer tableaux, and qudit-to-qubit mappings.

Findings

Galois qudits of dimension 2^s are equivalent to collections of s qubits.

Formal definitions and properties of Galois qudits and their Clifford hierarchies.

Description of quantum Reed-Solomon codes using Galois qudits.

Abstract

Galois qudits are $q$-dimensional quantum systems whose choice of Pauli group encodes the arithmetic of some finite field $\mathbb{F}_q$. They differ from the more familiar modular qudit, which are the same quantum system but whose choice of Pauli group are the clock and shift operators, which encode the arithmetic of integer addition and multiplication modulo $q$. Galois qudits are a useful mathematical construct that allow us to leverage the mathematical tools that are native to the larger qudit while only physically building smaller qudits. In particular, a Galois qudit of dimension $q = 2^s$ is exactly the same thing as a collection of $s$ qubits, not only in its Hilbert space, but also in its Pauli group, and Clifford hierarchy. This formalism has found a lot of utility recently in constructing quantum error-correcting codes over qubits with useful properties. In this review, we build on existing literature to collect and formalise facts and proofs about Galois qudits over binary extension fields. We define them and their Clifford hierarchies, describe what it means to measure their Pauli operators, describe their stabiliser tableaux, formally define qudit-to-qubit mappings, and finally describe quantum Reed-Solomon codes.