Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 4

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]

Let $x\in \{00,01,10,11\}$ and let $f(x)$ be one only at the marked input. A three-qubit implementation of Grover's algorithm starts with $|00⟩|1⟩$, applies a Hadamard gate to each qubit, queries the reversible quantum oracle $U_f:|x,y⟩\mapsto |x,y\oplus f(x)⟩$, applies the Grover diffusion operator $D=2|s⟩⟨s|-I$ to the first two qubits, and measures those two qubits in the computational basis. Here $|s⟩=\frac{1}{2}{\sum }_x|x⟩$. The full three-wire quantum circuit is the composition

Here $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Since $X|-⟩=-|-⟩$, phase kickback gives $U_f(|x⟩|-⟩)=(-1{)}^{f(x)}|x⟩|-⟩$. The third qubit remains separable and unchanged. A direct marked-state phase oracle $O_f|x⟩=(-1{)}^{f(x)}|x⟩$ therefore replaces the three-qubit query using only two qubits:For the explicit choice $f(x)=1$ only at $x=11$, $O_f$ is a Controlled-Z gate. Immediately after it,The mean of these four probability amplitudes is $\overline{a}=1/4$. The Grover diffusion operator maps each probability amplitude $a_x$ to $2\overline{a}-a_x$. The three unmarked probability amplitudes become zero and the marked one becomes one. A single query consequently finds $11$ with probability one.

A gate-level implementation of $D$ is $H^{\otimes 2}(2|00⟩⟨00|-I)H^{\otimes 2}$. Up to an irrelevant global phase, it is $H^{\otimes 2}{X}^{\otimes 2}\mathrm{CZ}{X}^{\otimes 2}{H}^{\otimes 2}$. For any other chosen marked word, conjugate the Controlled-Z gate by Pauli X gates on precisely the positions where the marked word has a zero.

In liquid-state NMR quantum computing, use two distinguishable spin-$1/2$ atomic nuclei in one molecule as the two qubits. Their different resonance frequencies permit selective radiofrequency pulses for one-qubit rotations. In a suitable rotating frame, refocusing removes unwanted one-spin terms while the NMR scalar coupling suppliesAssuming $J>0$, free evolution for $t=1/(2J)$ implements $U_J=e^{-i\pi Z_1{Z}_2/4}$. Define $R_z(\theta )=e^{-i\theta Z/2}$. ThenThis equality follows directly by evaluating the four diagonal entries: the first three are $+1$ and the $11$ entry is $-1$. Negative $J$ can be handled by refocusing or reversing the effective interaction sign. The global phase can be ignored.

Thus the marked-state phase oracle for $11$ requires scalar-coupling evolution and calibrated one-spin rotations. For a marked word $b_1{b}_2$, conjugate by $X_1^{1-b_1}{X}_2^{1-b_2}$, realized as selective $\pi$ pulses. The Grover diffusion operator uses the same entangling operation with additional one-qubit pulses. Liquid-state NMR quantum computing usually reads an ensemble signal and often prepares an effective pseudo-pure deviation from a large mixed background; that reproduces this small quantum algorithm's signal without providing a scalable source of pure initialized qubits.