Exercises: Chapter 27 — Trapped Ion Qubits: Individual Atoms Manipulated by Lasers — IonQ, Quantinuum, and the Highest-Fidelity Qubits

Exercise 27.1: Paul Trap Stability

For a linear Paul trap with $r_0 = 500\ \mu\text{m}$, $\Omega_{\text{RF}}/2\pi = 30\ \text{MHz}$, and $V_0 = 500\ \text{V}$, calculate the stability parameters $q_x$ and $a_x$ for a $^{171}\text{Yb}^+$ ion. Determine whether the ion is stably trapped. If the RF amplitude is increased to $V_0 = 1000\ \text{V}$, what is the new radial secular frequency $\omega_r$ in the pseudopotential approximation? How does this affect the Lamb–Dicke parameter and gate speed?

Exercise 27.2: Mølmer–Sørensen Gate Derivation

Starting from the MS interaction Hamiltonian $\hat{H}_{\text{MS}} = \hbar\Omega_{\text{MS}}\hat{S}_x(\hat{a}e^{-i\delta t} + \hat{a}^\dagger e^{i\delta t})$, derive the time-evolution operator using the Magnus expansion. Show that at $t_g = 2\pi/\delta$, the motion disentangles. Compute the geometric phase acquired and verify that for $\Phi = \pi/2$, the gate is locally equivalent to a CNOT. What happens if the detuning $\delta$ fluctuates by $\pm 1\%$? Estimate the resulting gate infidelity.

Exercise 27.3: Doppler Cooling Limit

Calculate the Doppler cooling limit temperature and the corresponding mean motional occupation $\bar{n}$ for $^{40}\text{Ca}^+$ (cooling transition at 397 nm, $\Gamma/2\pi = 22\ \text{MHz}$) in a trap with $\omega_z/2\pi = 1\ \text{MHz}$. How many sideband cooling cycles are needed to reach $\bar{n} < 0.1$ if each cycle removes one quantum of motion? What is the total sideband cooling time if each cycle takes $1/\Omega_{\text{red}} \approx 10\ \mu\text{s}$?

Exercise 27.4: All-to-All vs. Nearest-Neighbor

Compare the circuit depth for implementing an $N$-qubit Quantum Fourier Transform (QFT) on (a) a trapped-ion processor with all-to-all connectivity and (b) a superconducting processor with nearest-neighbor connectivity on a 2D grid. For $N = 10$, estimate the ratio of circuit depths. What does this imply for the types of algorithms best suited to each platform? Consider also the effect of gate fidelity: if trapped-ion 2Q gates have 99.9% fidelity and superconducting 2Q gates have 99.7% fidelity, which platform produces higher-fidelity QFT results for $N = 10$?

Exercise 27.5: Photonic Interconnect Design

Design a photonic interconnect between two trapped-ion modules. Specify the required optical system (lens NA, fiber coupling efficiency, detector efficiency) to achieve an entanglement rate of 10 s$^{-1}$. Assume an attempt rate of 100 kHz and a photon collection efficiency of 10% from the ion. What is the fidelity of the resulting entangled pair if the single-photon detection has a dark count rate of 10 Hz? How does the entanglement rate scale with the number of modules in a network?

Exercise 27.6: Ion Chain Normal Modes

For 4 ions in a linear trap with $\omega_z/2\pi = 1\ \text{MHz}$, compute all 4 axial normal mode frequencies and eigenvectors. Show that the COM mode has uniform participation ($b_i = 1/\sqrt{4}$ for all $i$) and that the highest non-COM mode (the "stretch" mode) has alternating signs. Calculate the Lamb–Dicke parameter for each mode and discuss which mode is optimal for MS gates.

Exercise 27.7: Sympathetic Cooling Analysis

In a QCCD architecture, $^{171}\text{Yb}^+$ qubit ions are co-trapped with $^{138}\text{Ba}^+$ coolant ions. The coolant ions are continuously laser-cooled on the $^2S_{1/2} \leftrightarrow ^2P_{1/2}$ transition at 493 nm. Estimate the cooling rate for a Yb$^+$ ion with initial $\bar{n} = 10$ in a 2 MHz axial mode, given that the Ba$^+$ cooling rate is $\Gamma_{\text{cool}}/2\pi = 15\ \text{MHz}$ and the mode participation of Ba$^+$ is $b_{\text{Ba}}^2 = 0.3$. How long does it take to reach $\bar{n} < 0.05$?

Exercise 27.8: Gate Error Budget

A trapped-ion MS gate has the following error sources: (a) spontaneous emission: $2 \times 10^{-4}$, (b) motional heating: $5 \times 10^{-4}$, (c) laser intensity noise: $1 \times 10^{-4}$, (d) addressing errors: $3 \times 10^{-4}$, (e) decoherence during the gate: $1 \times 10^{-4}$. Calculate the total gate infidelity assuming these errors are independent. If you could improve one error source by a factor of 5, which one would you choose? What is the resulting total infidelity?

Exercise 27.9: QCCD Shuttling Waveform

Design a shuttling waveform to move an ion from position $x = 0$ to $x = 100\ \mu\text{m}$ in 50 $\mu\text{s}$ while keeping the motional excitation below $\Delta\bar{n} = 0.1$. Use a sinusoidal velocity profile: $v(t) = v_0 \sin(\pi t / T)$. Calculate $v_0$ from the distance constraint, then compute the residual motional excitation. Is the 50 $\mu\text{s}$ shuttling time sufficient, or do you need a longer transfer time?

Exercise 27.10: Comparison with Superconducting Qubits

A trapped-ion processor has 20 qubits with all-to-all connectivity, 99.9% two-qubit gate fidelity, and 200 $\mu\text{s}$ gate time. A superconducting processor has 100 qubits with nearest-neighbor connectivity on a 2D grid, 99.7% two-qubit gate fidelity, and 50 ns gate time. For implementing a 10-qubit QFT (which requires 45 two-qubit gates on nearest-neighbor but only 9 on all-to-all), calculate the total circuit execution time and estimated success probability for each platform. Which platform would you choose and why?