ControlledU

ControlledU, often written as CU, is a two-qubit quantum gate that applies a unitary operation U to a target qubit conditioned on the control qubit being in the state 1. When the control qubit is 0, the target qubit is left unchanged. Formally, CU = |0⟩⟨0| ⊗ I + |1⟩⟨1| ⊗ U, where I is the identity on the target qubit.

In the standard computational basis {|00⟩, |01⟩, |10⟩, |11⟩}, CU has a block-diagonal form: CU = diag(I, U). Equivalently, it is the 4×4 matrix that acts as the identity on the subspace where the control is 0 and applies U on the subspace where the control is 1.

Special cases illustrate the concept: if U = X (the Pauli-X gate), CU becomes the CNOT gate; if U = Z, CU is the controlled-Z gate. If U = I, CU is the identity on both qubits. The definition generalizes to higher-dimensional systems and to gates with multiple control qubits, where the operation U is applied when all controls are in the 1 state.

Applications and significance: controlled-U gates are fundamental for constructing conditional operations in quantum circuits. They are essential in many algorithms, including phase estimation, where a sequence of controlled-U gates encodes eigenphases, and in oracle constructions for search algorithms. They also enable the implementation of powers U^k by repeatedly applying U with the same control.

Implementation: In practice, CU can be synthesized from native single-qubit gates and two-qubit entangling gates (such as CNOT or other hardware-native gates) depending on the quantum architecture. Some platforms provide CU as a primitive, while others require decomposition into a sequence of basic gates.

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