Controlled-phase gate (CP)

Definition

$$\mathrm{CP}(\lambda) = \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&e^{i\lambda} \end{pmatrix}$$

Like $\mathrm{CZ}$, this matrix is symmetric under exchanging the two qubits, so "control" and "target" are interchangeable labels rather than a physical asymmetry.

Action on basis states

$$\mathrm{CP}(\lambda)|c,t\rangle = e^{i\lambda\, c\cdot t}|c,t\rangle$$

Only the $|11\rangle$ amplitude picks up a phase of $e^{i\lambda}$; every other computational-basis input is unaffected.

Key identities

• $\mathrm{CP}(\pi) = \mathrm{CZ}$: the controlled-$Z$ gate is the $\lambda=\pi$ special case of the controlled-phase family.
• $\mathrm{CP}(\lambda_1)\,\mathrm{CP}(\lambda_2) = \mathrm{CP}(\lambda_1+\lambda_2)$: controlled phases with the same qubit pair compose additively.
• General controlled-$U$ construction (Barenco et al. 1995): any single-qubit controlled-$U$ can be built from two CNOTs and three single-qubit gates $A,B,C$ satisfying $ABC = I$ and $AXBXC = U$ up to phase; $\mathrm{CP}(\lambda)$ is the special case where $U = P(\lambda)$ is already diagonal, so no CNOTs are needed at all — it is native.
• The quantum Fourier transform on $n$ qubits applies $\mathrm{CP}(\pi/2^{k})$ between qubit pairs at increasing separation $k$, making $\mathrm{CP}$ the single most-used two-qubit gate in QFT-based algorithms (phase estimation, Shor's algorithm, HHL).