Two-qubit gate ($U\in U(4)$)

A two-qubit gate is a unitary operator on two qubit lines, extended by the identity on the others. It may be entangling, as a CNOT gate is, or a product of local operations. In a Hamiltonian simulation model allowing arbitrary two-qubit gates, the exponential of a Hermitian term supported on two lines counts as one such gate. Compiling it into a fixed discrete universal quantum gate set introduces a separate approximation cost.

A two-qubit gate in which an interaction shifts the energy of a selected joint computational basis state and thereby accumulates a relative quantum phase. With Hamiltonian operator $H=\Omega |01⟩⟨01|$, where $\Omega$ has units of energy, an interaction time $\tau$ gives $U_{\varphi }=\mathrm{diag}(1,e^{-i\varphi },1,1)$ with $\varphi =\Omega \tau /\hslash$. The selected basis state is part of the convention: this gate differs from the usual Controlled-Z gate at $\varphi =\pi$. Conjugation by a Pauli X gate on the first qubit gives $(X\otimes I)U_{\pi }(X\otimes I)=\mathrm{CZ}$, because the conjugation interchanges $|01⟩$ and $|11⟩$.

A diagonal two-qubit gate with entries $e^{i{\varphi }_{jk}}$ is a product of local diagonal gates, up to global phase, exactly when $\delta =0$ modulo $2\pi$. This follows by factoring the two-by-two array of phases; its determinant vanishes precisely in that case. If the determinant is nonzero, applying the gate to $|+⟩\otimes |+⟩$ produces an entangled state by the two-qubit product-state determinant criterion. In particular $\mathrm{diag}(-1,1,1,-1)=-Z\otimes Z$ is local, whereas $\mathrm{diag}(1,1,1,-1)$ is the entangling Controlled-Z gate.

Every two-qubit gate has the displayed form with local elements of the special unitary group and real interaction parameters. The three Pauli-product generators commute, so the central exponential is the product of three interactions. Local basis changes turn an Ising coupling into either of the other Pauli-product couplings. A fixed entangling controlled phase together with arbitrary local rotations can also synthesize each variable interaction, for example through two CNOT gates enclosing a variable $z$ rotation.