Three-Qubit State Preparation: Classification and Explicit Circuits

TL;DR

This paper introduces a deterministic method for preparing any three-qubit pure state by classifying states into five types and providing explicit, practical circuit templates based on entanglement structure and Schmidt decomposition.

Contribution

It offers a systematic, explicit circuit construction for three-qubit states, improving practicality and efficiency over previous methods by using only local CNOT gates and providing clear criteria for state classification.

Findings

Explicit circuit templates for all three-qubit state types

Gate parameters derived from Schmidt decomposition

Practical circuits with reduced gate counts and depth

Abstract

We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a $1|2$ bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.