Applies a configurable phase rotation to the |11⟩ state.
CP is the parameterized generalization of CZ. Where CZ always applies a −1 phase to |11⟩, CP lets you choose any phase angle θ. At θ = π, CP becomes CZ. At smaller angles, it applies more subtle phase rotations. This fine-grained phase control is exactly what appears in the Quantum Fourier Transform, where each qubit pair gets a CP gate with a different angle, building up the precise phase relationships needed for the transform.
CP(θ) = diag(1, 1, 1, exp(iθ)). It applies a phase of exp(iθ) to the |11⟩ component only. Like CZ, CP is symmetric between the two qubits. The gate is unitary and its inverse is CP(−θ). The QFT on n qubits uses CP gates with angles π/2^k for k = 1, ..., n−1.
Joint phase structure changes by angle θ.
At θ = π this becomes the CZ gate.
Conditional phase rotation by θ. Symmetric between qubits. At θ = π, becomes CZ.
CP might look like a minor variation of CZ, but the ability to choose arbitrary angles is what makes algorithms like the QFT work. The precision of these angles directly determines the resolution of quantum phase estimation.
CP is the engine of the Quantum Fourier Transform, which underlies Shor’s factoring algorithm and quantum phase estimation. Understanding CP means understanding how quantum computers extract frequency and period information exponentially faster than classical computers.
In the Quantum Fourier Transform, why do the CP angles decrease as π/2^k?