Three-Qubit State Preparation: Classification and Explicit Circuits

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.

Figures (5)

• Figure 1: General scheme Murta2023 for preparing an arbitrary three-qubit pure state $\ket{\psi_3}_{ABC}$ via the Schmidt decomposition with respect to the $A|BC$ bipartition in Eq. (\ref{['eq:Psi3SD']}). The unitaries $U_A$ and $U_{BC}$ are defined in Eqs. (\ref{['eq:Ua']}) and (\ref{['eq:Ubc']}). To instantiate this scheme, we classify three-qubit pure states into five entanglement types in Sec. \ref{['sec:classify']}. If the target state is fully separable or biseparable across the $A|BC$ bipartition, only $U_A$ and $U_{BC}$ are required, whereas SS-, SE-, and EE-type $A|BC$-entangled states require the full circuit including the $R_y$ rotation and the CNOT gate.
• Figure 2: Circuit for preparing an arbitrary single-qubit state $\ket{\psi_1}$ as in Eq. (\ref{['eq:Psi1']}). This circuit serves as a building block for preparing biseparable three-qubit pure states. Since a global phase has no observable physical effect, $\Phi$ is introduced only to indicate mathematically exact state preparation; it is therefore drawn with a dashed outline and can be omitted in practice. The same building blocks also suffice to prepare fully separable three-qubit pure states.
• Figure 3: Circuit for preparing an arbitrary two-qubit state $\ket{\psi_2}$ as in Eq. (\ref{['eq:Psi2']}), equivalently expressed in the Schmidt form of Eq. (\ref{['eq:Psi2SD']}). This circuit serves as a building block for preparing biseparable three-qubit pure states. We assume that the two-qubit state is entangled with respect to the $B|C$ bipartition. The circuit constructions are divided into two cases according to the diagonal coefficient $\mathcal{D}$ in Eq. (\ref{['eq:Diagonal']}): (a) generic case $\mathcal{D}\neq 0$, where, to simplify the gate parametrization, the single-qubit unitaries $V_B$ and $V_C$ are expressed in terms of the Schmidt coefficients $\tau_j$ and the auxiliary quantities $B_j$, $E_j$, and $F_j$ defined in Eq. (\ref{['eq:SDNonDiagonal']}); (b) special case $\mathcal{D}=0$, where the simpler Schmidt decomposition in Eq. (\ref{['eq:SDDiagonal']}) allows the gate parameters to be represented directly in terms of the amplitudes $d_j$ in Eq. (\ref{['eq:Psi2']}).
• Figure 4: Circuits for preparing an $A|BC$-entangled three-qubit state $\ket{\psi_3}$. Based on the Schmidt decomposition in Eq. (\ref{['eq:Psi3SD']}), we construct state-preparation circuits for the SS-, SE-, and EE-type families. (a) Circuit for preparing SS-type states. The angles $\theta$ and $\phi_j$ are given in Eq. (\ref{['eq:Mapping3']}), and the final single-qubit unitaries $U_A$, $W_B$, and $W_C$ are defined in Eqs. (\ref{['eq:Ua']}) and (\ref{['eq:WbWc']}). (b) Circuit for preparing SE-type states. The circuit parameters $\theta_j$ and $\phi_j$ are specified in Eq. (\ref{['eq:SEVector']}), and $U_A$, $W_B$, and $W_C$ are defined in Eqs. (\ref{['eq:Ua']}) and (\ref{['eq:SEWbSEWc']}). (c) Circuit for preparing EE-type states. The phase $\phi_0$ and the parameters $\kappa_j$ and $\sigma_j$ are specified in Eq. (\ref{['eq:EEVector']}) and in Eqs. (\ref{['eq:Kappa0']})--(\ref{['eq:Sigma1']}), respectively. The final single-qubit unitaries $U_A$, $W_B$, and $W_C$ are defined in Eqs. (\ref{['eq:Ua']}) and (\ref{['eq:EEWbWc']}).
• Figure 5: Explicit circuits for preparing the $\mathrm{R}_1$--$\mathrm{R}_4$ classes. (a) Circuit for the $\mathrm{R}_1$ class defined in Eq. (\ref{['eq:R1']}). (b) Circuit for the $\mathrm{R}_2$ class. The phases $\varphi_j$ appear in Eq. (\ref{['eq:R2']}), and the rotation angles $\theta_j$ are given in Eq. (\ref{['eq:R2Theta']}). (c) Circuit for the $\mathrm{R}_3$ class. The gate parameters are determined by the state parameter $\varphi_2$ in Eq. (\ref{['eq:R3']}) and by $\theta_j$ and $\phi_j$ in Eq. (\ref{['eq:R3Phi']}). (d) Circuit for the $\mathrm{R}_4$ class. The state parameters $r_j$ and $\varphi_j$ in Eq. (\ref{['eq:R4']}) determine the gate parameters $\theta_j$ and $\phi_j$ in Eqs. (\ref{['eq:R4Theta0']})--(\ref{['eq:R4Phi2']}). By choosing appropriate parameter values, these circuits prepare the states listed in Table \ref{['tab:All']}.

Theorems & Definitions (2)

• Definition 1: type classification
• Proposition 2: $A|BC$ separability