Optimal preparation of graph states

TL;DR

This paper presents an optimal method for preparing graph states of up to 12 qubits using minimal controlled-Z gates and depth, leveraging local Clifford equivalence and entanglement invariants.

Contribution

It introduces a systematic approach to prepare graph states with minimal resources, extending classification and invariants for up to 12 qubits.

Findings

Optimal circuits with minimal controlled-Z gates

Optimal circuits with minimal preparation depth

Explicit gate sequences for various graph states

Abstract

We show how to prepare any graph state of up to 12 qubits with: (a) the minimum number of controlled-Z gates, and (b) the minimum preparation depth. We assume only one-qubit and controlled-Z gates. The method exploits the fact that any graph state belongs to an equivalence class under local Clifford operations. We extend up to 12 qubits the classification of graph states according to their entanglement properties, and identify each class using only a reduced set of invariants. For any state, we provide a circuit with both properties (a) and (b), if it does exist, or, if it does not, one circuit with property (a) and one with property (b), including the explicit one-qubit gates needed.