CNOT decomposer

The CNOT decomposer (CNOTDecomposer) 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) CNOT gates and single-qubit gates. It decomposes the CR, CRk and CZ controlled two-qubit gates.

The SWAP gate is not decomposed by the CNOT decomposer

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

The CNOT gate is, in general, not part of the primitive gate set of a target backend. The CZ gate is more likely to be part of the primitive gate set. After using the CNOT decomposer one could use the CNOT to CZ decomposer (CNOT2CZDecomposer) or, instead of using the CNOT decomposer all together, simply decompose the two-qubit controlled gates using the CZ decomposer.

The algorithm for the CNOT decomposer is based on 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,

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

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 ZYZ decomposer (ZYZDecomposer), whereby the arbitary unitary \(U\) is written as,

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

which can be rearranged such that,

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

Decomposition using a single CNOT

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

\[ U = R_z(\theta_1) \cdot R_z(\theta_0) \cdot R_z(\theta_1) \cdot X, \]

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