7. Two-qubits Gates

When two gates \(A\) and \(B\) are applied to qubits \(q_0\) and \(q_1\), respectively, we can write the combined gates as \(C = A \otimes B\). We do not call it 2-qubit gate. In fact, \(A\) and \(B\) are not necessarily applied to the qubits at the same time. A 2-quibit gate cannot be expressed as a tensor product of two 1-qubit gates and it must act simultaneously on two qubits.

Commonly used 2-qubit gates are controoled gates, which are actually conditional 1-qubit gates. A 1-qubit gate such as X is applied to \(q_1\) only when \(q_0\) is in \(|1\rangle\). Otherwise, no action is taken.
There are other kind of 2-qubit gates such as Swap gate.

Most of common 2-qubit gates are spectial cases of more general canonical gates. Many models used in condensed matter physics are directly expressed by canonical gates, understandin the relation between canonical gates and common 2-qubit gates are particluarly important.

Generaic Name	Qiskit Class Name	Qiskit Ciruit Name	Symbols
Controlled X	CXGate	cx or cnot	CX or CNOT
Controlled Y	CYGate	cy	CY
Controlled Z	CZGate	cz	CZ
Controlled Hadamard	CHGate	ch	CH
Controlled SX	CSXGate	csx	CSX
Controlled Phase	CPhaseGate	cp	CP
Controlled Rotaion X	CRXGate	crx	CR\(_x\)
Controlled Rotaion Y	CRYGate	cry	CR\(_y\)
Controlled Rotaion Z	CRZGate	crz	CR\(_z\)
Controlled Unitary 1	CU1Gate	cu1	CU\(_1\)
Controlled Unitary 3	CU3Gate	cu3	CU\(_3\)
Controlled Unitary	CUGate	cu	CU

Others

• Swap

• H\(_2\)

Canonical Gates

Can\((t_x,t_y,t_z)=\exp\left[-i\frac{\pi}{2}\left( t_x X\otimes X + t_y Y\otimes Y + t_z \otimes ZZ\right) \right]\)