Five-qubit states generated by Clifford gates

TL;DR

This paper systematically classifies all 5-qubit Clifford states, analyzing their structure and the effect of controlled-Z gates, providing a detailed orbit partition and visual representations for understanding their transformations.

Contribution

It provides the first exhaustive classification of 5-qubit Clifford states and their orbits under local Clifford operations, including the action of controlled-Z gates.

Findings

Identified 19,388,160 5-qubit Clifford states.

Partitioned states into 93 orbits under local Clifford equivalence.

Presented diagrams and tables of CZ gate actions on orbits.

Abstract

The Clifford group is the set of gates generated by controlled-Z gates, the phase gate and the Hadamard gate. We will say that a n-qubit state is a Clifford state if it can be prepared using Clifford gates. These states are known as the stabilizer states and they arise in quantum error correction. In this paper we study the set of all 5-qubit Clifford states. By using an exhaustive method we start by confirming that there are 19388160 states. The main goal of the paper is to understand the action of the controlled-Z gates action on the 5-qubit states. With this goal in mind, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We show that there are 93 orbits, and we label each orbit in such a way that it is easy to see the effect of the controlled-Z gates. Diagrams and tables explaining the action of the CZ gates on all the orbits are presented in the paper. A similar work is done for the real Clifford 5-qubits states, this is, for states that can be prepared with Controlled-Z gates, the Z gate and the Hadamard gate.