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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.

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.

Apply one CNOT gate from each input qubit to the same target. Their actions commute because controls remain unchanged and each contribution is added modulo two. This reversible circuit computes the parity bit with exactly one two-qubit gate per input, and its inverse is the same circuit. Unlike measuring the inputs to compute parity classically, the unitary preserves quantum superposition and can be uncomputed after phase application.

Without interaction, the two-qubit Hamiltonian operator is $H_a\otimes I+I\otimes H_b$, so its evolution factors as $U_a\otimes U_b$. It maps each product state to another product state, and cannot implement an entangling two-qubit gate such as a CNOT gate.

In an ion trap, the Coulomb interaction couples the ions' displacements into collective vibrational normal modes. The qubit states are internal atomic states; lasers coupling internal excitation to creation or annihilation of a shared vibrational quantum allow that motion to carry conditional information between ions. Thus the physical mediator is a collective motional normal mode of the Coulomb-coupled chain, rather than a direct resonant exchange between isolated internal qubit transitions. Ground-state cooling and resolved motional sidebands support the ideal pulse model used here.

Track the $-i$ factors only when the corresponding transition is addressed. They are common within the addressed two-state subspace, but become physically relevant relative phases between computational basis states. Using a semicolon to separate the shared motional number state, the complete paths areThe middle arrow comprises both Y pulses. In the last row those individual steps are $-i|ge;1⟩\to -|gf;0⟩\to +i|ge;1⟩$. The two X pulses contribute $(-i{)}^2=-1$ for either initially excited control; the Y pair supplies a second minus sign only when the second ion is also excited.

The shared motion returns to its ground state for all four inputs, and the auxiliary internal state is empty. Quantum superposition therefore gives the two-qubit gateDiscarding the X-pulse factors would incorrectly identify $G$ with a Controlled-Z gate itself.

The five DiVincenzo criteria are: a scalable collection of well-characterized qubits; preparation in a known fiducial quantum state; relevant coherence times long compared with quantum gate times; a universal quantum gate set; and qubit-specific quantum measurement. The criterion is useful control of the whole computation, including errors, rather than merely observing two resolved levels.^[ref]

For trapped ions, the implementation is:

For trapped neutral atoms, the corresponding implementation is:

Thus neither platform is forbidden from meeting a criterion in principle. Weak neutral-ground-state interactions make entangling control harder, but claiming that trapped neutral atoms cannot implement a universal quantum gate set would be false. Gate-model quantum algorithms using Rydberg blockade had already been demonstrated in a programmable neutral-atom processor by 2022.^[ref]

Write $X={\sigma }_x$, $Y={\sigma }_y$, $Z={\sigma }_z$. The specified diagonal two-qubit gate has the exact factorizationThus one exact implementation is to set the controllable Ising coupling to zero and perform the two local rotations about the z-axis, synthesized from the available $x,y$ pulses as in part (c). Alternatively, turn off the local fields and choose the interaction area $\int J_{12}(t)dt=\pi /2$. Thenwhich implements the same physical gate up to global phase. The printed gate is local, not entangling; the pulse construction does not change that fact.

The printed upper summation limit introduces $Z_n$ although only qubits $0,\dots ,n-1$ were defined. Literally the final term is undefined. Use the natural open-chain repairThis preserves the stated $n$-qubit system and agrees with the supplied sum-of-squares hint. If a cyclic convention $Z_n=Z_0$ was intended instead, it must be stated; the same argument works for its $n$ terms when $n\ge 2$. For $n=1$, the repaired open chain is empty and the target is the identity.

Here is a product-formula Hamiltonian simulation using exactly the two supplied lemmas. Let $L=n-1$ and choose an integer $r\ge L$. Every $h_i$ is a norm-one Hermitian matrix. One time slice is the product of two-qubit gatesTo compare a partial product with $e^{i(h_1+\cdots +h_j)/r}$, first propagate the previous error through the next unitary gate, which preserves the spectral norm, and then use Lemma A with $A=-(h_1+\cdots +h_{j-1})/r$, $B=-h_j/r$. Both norms are at most $j/r\le 1$. Its new error is at most $c(j/r{)}^2$ for a universal constant $c$. For an explicit choice, $c=4$ follows from the unitary Taylor bounds $\Vert e^{iA}-I-iA\Vert \le \Vert A{\Vert }^2/2$ and $\Vert e^{iA}-I\Vert \le \Vert A\Vert$. Induction and the triangle inequality therefore giveLemma B, the unitary product telescoping bound, now compares the $r$ repeated slices with $(e^{i{H}_{\mathrm{open}}/r}{)}^r=e^{i{H}_{\mathrm{open}}}$:Take, for example, $r=\max (1,L,\lceil 2c{\sum }_{j=2}^L{j}^2/ϵ\rceil )$. If the sum is zero the product is already exact; otherwise its error is at most $ϵ/2<ϵ$. There are $Lr$ two-qubit gates. This explicit lemma-based construction has fourth-degree dependence on $n$ for fixed precision:This is a sufficient polynomial, not an optimality claim. The polynomial-degree statement treats $ϵ$ as fixed; the inverse-precision dependence is displayed separately. No first-order term error is accumulated without the required repeated-slice factor.

The Pauli Z gates give eigenvalue $(-1{)}^{{\sum }_j{x}_j}$ on $|x⟩$. Use parity computation by CNOT gates to put the parity bit in a zero ancilla, apply $e^{itZ}$ to it, and uncompute. The data acquire exactly $e^{it(-1{)}^{{\sum }_j{x}_j}}$, and the ancillary line returns to zero. This compute-phase-uncompute construction needs $2n+1$ one- and two-qubit gates. Accumulating parity into the last data line instead uses $2n-1$ gates without an extra ancilla. Neither implementation drops the global phase.

Rydberg blockade suppresses simultaneous excitation of nearby chemical atoms into strongly interacting Rydberg atom states. If the interaction shifts the doubly excited level by much more than the excitation linewidth and coherent drive scale, a pulse resonant for one isolated chemical atom becomes off-resonant when a neighbour is already excited. The resulting conditional dynamics can implement an entangling two-qubit gate for neutral-atom qubits.