Exercises: Chapter 34 — The Quantum Future: Fault Tolerance, Quantum Networks, Quantum Internet, and What Comes After NISQ

Exercise 34.1 — Quantum Repeater Rate Analysis. A quantum repeater chain of length $L$ has $N$ segments, each of length $L_0 = L/N$. The entanglement generation probability per segment is $p = e^{-L_0 / L_{\text{att}}}$ where $L_{\text{att}} = 22$ km (fiber attenuation length). The entanglement swapping success probability is $q = 0.9$. Derive the end-to-end entanglement generation rate as a function of $N$. Find the optimal $N$ for $L = 1,000$ km. Compare the rate with and without repeaters.

Exercise 34.2 — Barren Plateau Detection. Implement a function that computes the variance of the gradient of a parameterized quantum circuit's cost function. For a hardware-efficient ansatz with increasing depth, plot the gradient variance as a function of the number of qubits. At what depth does the variance drop below $10^{-6}$ for 4, 8, and 12 qubits?

Exercise 34.3 — Entanglement Swapping Simulation. Write a Qiskit simulation of entanglement swapping between three nodes (A, B, C). Generate an entangled pair between A-B and B-C. Perform a Bell-state measurement on B's two qubits. Verify that A and C become entangled. Compute the fidelity of the final A-C entangled state as a function of the initial entanglement fidelity.

Exercise 34.4 — Post-Quantum Migration Planning. Research the NIST post-quantum cryptography standards (finalized 2024). Write a one-page migration plan for a hypothetical e-commerce company with 10 million users, outlining the steps, timeline, and risks of transitioning from RSA-2048 to a post-quantum scheme.

Exercise 34.5 — Your Quantum Roadmap. Write a personal 5-year plan for your involvement in quantum computing. Identify: (a) your primary area of interest (algorithms, hardware, software, applications), (b) three specific skills you will develop, (c) two open-source projects you will contribute to, (d) one conference you will attend, and (e) a project you will build to demonstrate your skills.

Exercise 34.6 — Fault Tolerance Resource Estimation. Calculate the number of physical qubits required for a fault-tolerant quantum computer that can run Shor's algorithm for RSA-2048, using: (a) surface code with $p = 10^{-3}$ and $d = 27$, (b) surface code with $p = 10^{-4}$ and $d = 15$, (c) a hypothetical topological code with $p = 10^{-6}$ and $d = 7$. Compare the total qubit counts and discuss the implications for hardware development timelines.

Exercise 34.7 — Quantum Sensor Design. Design a quantum magnetometer using NV centers in diamond for biomedical imaging (magnetoencephalography). Calculate: (a) the required sensitivity to detect neural currents (~1 pT at 1 kHz bandwidth), (b) the number of NV centers needed at the SQL, (c) the improvement with Heisenberg-limited sensing, and (d) the technical challenges (laser power, readout fidelity, ensemble inhomogeneity).

Exercise 34.8 — Quantum Internet Protocol Design. Design a simple quantum network protocol for distributing Bell pairs between three nodes (A, B, C) connected by 50 km fiber links. Calculate: (a) the raw Bell pair generation rate per link, (b) the fidelity after entanglement swapping, (c) the rate after purification, and (d) compare with direct transmission over 100 km.

Exercise 34.9 — Ethical Analysis of Quantum Technology. Write a 1,000-word essay analyzing one of the following ethical dimensions of quantum computing: (a) the "harvest now, decrypt later" threat and the responsibility of governments and corporations to deploy post-quantum cryptography, (b) the potential for quantum computing to exacerbate global inequality if access is limited to wealthy nations, or (c) the dual-use nature of quantum simulation (drug discovery vs. toxin design).

Exercise 34.10 — Quantum Startup Evaluation. Choose a quantum computing startup (from Chapter 31) and write a detailed evaluation that includes: (a) their technical approach and current capabilities, (b) their competitive position relative to IBM/Google/Microsoft, (c) their path to profitability, (d) the biggest risks to their success, and (e) a recommendation (invest, wait, or avoid) with justification.