Exercises: Chapter 28 — Photonic, Neutral Atom, and Other Approaches: The Diversity of Quantum Hardware and Why No One Has Won Yet

Exercise 28.1: KLM Gate Success Probability

The original KLM CZ gate succeeds with probability 1/16. Suppose you have a source that produces entangled photon pairs at a rate of 1 MHz. If you need to implement 100 CZ gates for a computation, and failed gates require restarting the entire circuit, what is the expected time to complete the computation? How does this change if the success probability is improved to 1/4? Discuss the implications for scalable photonic quantum computing.

Exercise 28.2: Rydberg Blockade Radius

For $^{87}\text{Rb}$ atoms excited to the $n = 70$ Rydberg state, the van der Waals coefficient is $C_6/h \approx 900\ \text{GHz}\cdot\mu\text{m}^6$. If the Rydberg Rabi frequency is $\Omega_R/2\pi = 2\ \text{MHz}$, calculate the blockade radius $R_b$. If atoms are placed on a square grid with spacing $a$, what is the maximum $a$ that still allows nearest-neighbor gates? How many atoms can be packed into a $100 \times 100\ \mu\text{m}^2$ area at this spacing?

Exercise 28.3: Exchange Gate Calibration

Two electron spin qubits in a silicon double quantum dot have an exchange coupling $J(\varepsilon) = J_0 \exp(-\varepsilon/\varepsilon_0)$ with $J_0/h = 1\ \text{GHz}$ and $\varepsilon_0 = 10\ \text{mV}$. What gate voltage $\varepsilon$ is needed to implement a $\sqrt{\text{SWAP}}$ gate in $t = 10\ \text{ns}$? If voltage noise is $\delta\varepsilon = 0.1\ \text{mV}$ RMS, estimate the gate infidelity due to exchange noise.

Exercise 28.4: Topological Protection

A pair of Majorana zero modes separated by distance $L$ has an energy splitting $\delta E \approx \Delta e^{-L/\xi}$, where $\Delta$ is the superconducting gap and $\xi$ is the coherence length. For $\Delta/h = 10\ \text{GHz}$, $\xi = 100\ \text{nm}$, and $L = 1\ \mu\text{m}$, calculate $\delta E$. What is the corresponding dephasing time $T_2^* \approx \hbar/\delta E$? Compare this to the transmon $T_2^*$ from Chapter 26. How does the dephasing time scale with $L$? What are the practical limits on $L$?

Exercise 28.5: Hardware Selection

You are tasked with choosing a quantum computing platform for the following applications. For each, recommend a platform and justify your choice based on the comparison table in Section 28.6: (a) Running Shor's algorithm to factor a 2048-bit RSA key (requires $\sim$10$^8$ physical qubits with error correction). (b) Simulating the electronic structure of a 50-atom molecule (requires high-fidelity gates and moderate qubit count). (c) Building a quantum repeater for a 500 km quantum network link. (d) Deploying a quantum computer in a standard office environment (no cryogenics).

Exercise 28.6: Boson Sampling Complexity

Implement a boson sampling simulation for 4 photons in 8 modes. Generate a random 8-mode interferometer unitary, compute the output probability distribution using matrix permanents, and verify that the distribution is uniform over all possible output configurations. What is the largest number of photons you can simulate on your computer? How does the runtime scale?

Exercise 28.7: NV Center Sensitivity

An NV center in diamond has $T_2 = 1\ \text{ms}$ with dynamical decoupling at room temperature. Calculate the magnetic field sensitivity per $\sqrt{\text{Hz}}$ for: (a) a single NV center, (b) an ensemble of $10^6$ NV centers (assuming independent measurements), and (c) a single NV center at cryogenic temperature with $T_2 = 10\ \text{ms}$. How does this compare with SQUID magnetometers ($\sim 1\ \text{fT}/\sqrt{\text{Hz}}$)?

Exercise 28.8: Rydberg Gate Error Budget

A Rydberg CZ gate has the following error sources: (a) finite blockade ratio: $5 \times 10^{-3}$, (b) spontaneous emission from the Rydberg state ($\tau_r \sim 100\ \mu\text{s}$, gate time $t_g \sim 1\ \mu\text{s}$): $\sim 10^{-2}$, (c) Doppler dephasing: $2 \times 10^{-3}$, (d) laser intensity noise: $1 \times 10^{-3}$, (e) addressing errors (off-target excitation): $3 \times 10^{-3}$. Calculate the total gate infidelity. Which error source is dominant? What is the single most impactful improvement?

Exercise 28.9: Photonic Loss Budget

A photonic quantum circuit requires 100 beamsplitters, each with 0.5% loss. What is the total transmission probability for a single photon traversing the entire circuit? If each photon must pass through the circuit with >1% probability, what is the maximum number of beamsplitters? How does this affect the scalability of linear optical quantum computing?

Exercise 28.10: Platform Comparison for Quantum Simulation

Compare the following platforms for simulating a 20-qubit Heisenberg model with nearest-neighbor interactions on a 1D chain: (a) superconducting qubits with nearest-neighbor connectivity, (b) trapped ions with all-to-all connectivity, and (c) neutral atoms with Rydberg blockade connectivity. Estimate the total circuit execution time and the expected fidelity for each platform, assuming typical parameters from the comparison table.