CZ decomposer

The CZ decomposer (CZDecomposer) is a two-qubit gate decomposer. When applied to a circuit, it decomposes every controlled two-qubit gate into a sequence of (at most 2) CZ gates and single-qubit gates. It decomposes the CR, CRk and CNOT controlled two-qubit gates.

The SWAP gate is not decomposed by the CZ decomposer

The SWAP gate is not a controlled two-qubit gate and can therefore not be decomposed using the CZ decomposer. To decompose a SWAP gate into either CZ or CNOT gates, use the predefined SWAP to CZ decomposer (SWAP2CZDecomposer) or SWAP to CNOT decomposer (SWAP2CNOTDecomposer).

The CZ gate is more likely to be part of the primitive gate set over the CNOT gate. It is therefore recommended to decompose two-qubit controlled gates using the CZ decomposer over the CNOT decomposer (CNOTDecomposer).

The algorithm for the CZ decomposer is derived from that of the CNOT decomposer (CNOTDecomposer), which in turn is based on the the ABC decomposition (not to be confused with the ABA decomposer). The procedure is described in Quantum Gates, section 7.5 by G. Crooks (2024).

The single-qubit unitary \(U\) of an arbitrary controlled-U gate, can be written as,

\[U = \text{e}^{i\alpha} A \cdot Z \cdot B \cdot Z \cdot C, \]

where \(\alpha\) denotes a global phase factor, and the single-qubit unitaries \(A\), \(B\), and \(C\), are chosen such that \(A \cdot B \cdot C = I\). In terms of the latter, the decomposition looks as illustrated below:

The ABC decomposition is made using the ABA decomposer, and in particular, the XYX decomposer (XYXDecomposer), whereby the arbitary unitary \(U\) is written as,

\[ U = \text{e}^{i\alpha} R_x(\theta_2) \cdot R_y(\theta_1) \cdot R_x(\theta_0), \]

which can be rearranged such that,

\[ \begin{align*} A& = R_x(\theta_2) \cdot R_y(\theta_1/2), \\ B& = R_y(-\theta_1/2) \cdot R_x(-\theta_0/2 - \theta_2/2), \\ C& = R_x(\theta_0/2 - \theta_2/2). \\ \end{align*} \]

Decomposition using a single CZ

In certain cases where the unitary \(U\) can be written as

\[ U = R_x(\theta_1) \cdot R_y(\theta_0) \cdot R_x(\theta_1) \cdot Z, \]

only a single CZ gate is required in the decomposition, according to Lemma 5.5 in Barenco et al. 1995.