Quantum logic gate ($U^{†}U=I$)

A unitary operation on a fixed number of qubits used as a component of a quantum circuit. Examples include the Hadamard gate and Controlled-Z gate. Measurements are described separately by quantum measurement; a gate is a reversible operation on its full input state.

A unitary gate on two quantum systems is entangling if it maps at least one pure state that is a product state to an entangled state. The controlled-NOT gate qualifies because it maps $|+⟩|0⟩$ to a Bell state. A gate that merely exchanges the two systems need not be a local unitary operation, but cannot create quantum entanglement from a product state.

For two $d$-dimensional unitary matrices, this normalized absolute trace lies in $[0,1]$ and equals one exactly when $V=e^{i\gamma }U$. The triangle inequality on the eigenphases of $U^{†}V$ proves both statements. It ignores an overall global phase and compares complete propagators. It is not the same definition as average gate fidelity; for unitary implementations that latter quantity is $(d{\mathcal{F}}^2+1)/(d+1)$.

A method that combines a prepared magic state, Clifford gates, and measurements with outcome-dependent corrections to implement a non-Clifford operation on a data register.

Apply a controlled-NOT gate from data $a|0⟩+b|1⟩$ to the resource $|A⟩$, then measure the resource in the standard basis. Outcome zero leaves data $T|\psi ⟩$; outcome one leaves $e^{i\pi /4}{T}^{†}|\psi ⟩$. Both outcomes have probability $1/2$. Correct outcome one with $S$, using $S{T}^{†}=T$, so the data has undergone the T gate irrespective of the measurement result. The resource is consumed, and its Bloch vector lies outside the stabilizer octahedron.

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.