Chapter 30 Exercises: The State of the Art — Where Quantum Physics Is Going

Part A: Conceptual Questions (⭐)

These questions test your understanding of the core ideas. No calculations required.

A.1 Explain the difference between "quantum supremacy" and "quantum advantage." Why has the physics community increasingly preferred the latter term? Give a concrete example illustrating why the distinction matters.

A.2 A technology journalist writes: "Google's quantum computer solved in 200 seconds a problem that would take a classical supercomputer 10,000 years. Quantum computers are now faster than classical computers." Identify at least three things wrong or misleading about this statement.

A.3 Why does quantum error correction require so many physical qubits per logical qubit? What is the fundamental difficulty that makes quantum error correction harder than classical error correction? (Hint: think about the no-cloning theorem.)

A.4 Explain why quantum sensing does not require quantum error correction, whereas quantum computing does. What is fundamentally different about the computational requirements of the two applications?

A.5 A colleague claims: "QKD is unbreakable because it is based on the laws of physics, not on mathematical assumptions." Is this claim fully accurate? Discuss the distinction between theoretical security and practical security, and give at least two examples of how real QKD systems can be attacked despite the theoretical security proof.

A.6 Feynman argued in 1981 that simulating quantum systems on classical computers is exponentially hard. Is this a proven mathematical theorem, or an empirical observation and conjecture? What complexity-theoretic result would be needed to make it rigorous?

A.7 Explain the difference between analog and digital quantum simulation. For each, give one example of a physical platform and one example of a problem it has been used to study. Under what circumstances would you prefer one approach over the other?

A.8 Why is the black hole information paradox a problem at the intersection of quantum mechanics and general relativity, rather than a problem within either theory alone? What principle of quantum mechanics does information loss violate?

A.9 A friend considering graduate school asks: "Should I go into quantum computing, or is the field overhyped?" Write a balanced, honest assessment of 200–300 words, acknowledging both the genuine promise and the legitimate concerns.

A.10 In the context of quantum gravity, explain what is meant by the statement "spacetime is built from entanglement." Which research program does this idea come from, and what is the key formula connecting entanglement entropy to geometry?

Part B: Applied Problems (⭐⭐)

These problems require quantitative reasoning and direct application of concepts from the chapter and prerequisites.

B.1: Quantum Advantage Scaling

Suppose a NISQ device can run a circuit of depth $d$ on $n$ qubits with total error rate per gate of $\epsilon$. The probability that the entire circuit runs without error is approximately $(1 - \epsilon)^{nd}$.

(a) For $n = 100$ qubits, $d = 100$ layers, and $\epsilon = 10^{-3}$, calculate the probability that the circuit runs error-free.

(b) For the same parameters but $\epsilon = 10^{-4}$, recalculate.

(c) What gate error rate $\epsilon$ is needed to achieve a 50% probability of error-free execution for a circuit with $n = 100$, $d = 1000$?

(d) Explain why this simple model underestimates the impact of errors in practice. What does error correction add to this picture?

B.2: Heisenberg Limit vs. Standard Quantum Limit

An atom interferometer uses $N = 10^6$ independent (unentangled) atoms to measure a phase shift $\phi$ due to gravity.

(a) What is the phase uncertainty $\Delta\phi$ at the standard quantum limit?

(b) If the atoms are entangled in an optimal state, what is the Heisenberg limit $\Delta\phi_{\text{HL}}$?

(c) Express the improvement factor $\Delta\phi_{\text{SQL}} / \Delta\phi_{\text{HL}}$ in terms of $N$, and evaluate it for $N = 10^6$.

(d) Explain why achieving the Heisenberg limit in practice is much harder than the simple formula suggests. What physical effects degrade the entanglement-based advantage?

B.3: QKD Key Rate Estimation

A fiber-based QKD system uses BB84 at a wavelength of 1550 nm (telecom band). The fiber attenuation is $\alpha = 0.2$ dB/km. Alice sends single photons at a rate of $R_0 = 10^9$ photons/second.

(a) What fraction of photons survive transmission through $L = 100$ km of fiber? (Use $\text{fraction} = 10^{-\alpha L / 10}$.)

(b) If the detector efficiency is $\eta_d = 0.1$ (10%) and the protocol efficiency is $\eta_p = 0.5$ (half the raw bits are discarded in basis reconciliation), estimate the raw key rate (detected bits per second) at $L = 100$ km.

(c) Repeat for $L = 200$ km and $L = 400$ km.

(d) At what distance does the raw key rate drop below 1 bit per second? Comment on the practical implications.

B.4: Quantum Simulation Hilbert Space

Consider the Hubbard model on an $L \times L$ square lattice at half-filling (one electron per site on average), with $N = L^2$ sites.

(a) Each site can be in one of four states: empty, spin-up, spin-down, or doubly occupied. Show that the total Hilbert space dimension is $4^N$.

(b) At half-filling, the constraint is that the number of spin-up electrons plus the number of spin-down electrons equals $N$. The dimension of this sector is $\left[\binom{N}{N/2}\right]^2$ (choosing which sites have spin-up, independently choosing which sites have spin-down). Calculate this for $N = 16$ ($4 \times 4$ lattice).

(c) Calculate the same for $N = 100$ ($10 \times 10$ lattice). Express your answer as a power of 10. (Use Stirling's approximation: $\ln \binom{N}{N/2} \approx N \ln 2 - \frac{1}{2}\ln(\pi N / 2)$.)

(d) A classical computer with 1 TB of RAM can store roughly $10^{11}$ double-precision complex numbers. What is the largest Hubbard model it could simulate exactly at half-filling (i.e., what is the largest $N$ for which the half-filling Hilbert space dimension fits in 1 TB)?

Part C: Analysis and Evaluation (⭐⭐⭐)

These problems require synthesis across multiple topics and critical evaluation.

C.1: Evaluating a Quantum Computing Claim

A startup announces that its 72-qubit quantum processor has achieved "quantum advantage" on a combinatorial optimization problem (Max-Cut on random graphs). The press release claims a "10x speedup over the best classical algorithm."

(a) List at least five questions you would want answered before accepting this claim.

(b) The company used QAOA at depth $p = 3$. Explain what this means and why the depth matters.

(c) The "best classical algorithm" they compared against was brute-force enumeration. Why is this comparison misleading? What classical algorithms should they have compared against?

(d) Suppose an independent group shows that a classical heuristic (e.g., simulated annealing) solves the same instances in less time than the quantum processor. Does this definitively disprove quantum advantage for this problem? Why or why not?

C.2: Quantum Repeater Design

You are designing a quantum repeater network to distribute entanglement over $L = 1000$ km.

(a) If the fiber attenuation is 0.2 dB/km and the entanglement distribution rate over a single 50-km segment is $R_{\text{seg}} = 10^4$ entangled pairs per second, estimate the end-to-end entanglement rate without repeaters.

(b) With quantum repeaters spaced every 50 km (20 segments), and assuming ideal entanglement swapping (no loss of fidelity), estimate the end-to-end rate. Assume the rate is limited by the slowest segment, and that each entanglement swapping operation has a success probability of 0.5.

(c) In practice, entanglement fidelity degrades with each swap. If each swap reduces the fidelity by a factor of 0.95, what is the end-to-end fidelity after 19 swaps? Is this above the threshold for useful QKD (typically $F > 0.9$)?

(d) What role does entanglement purification play in addressing the fidelity problem? At what cost in terms of rate?

C.3: The Societal Impact of Quantum Technologies

This is an essay question (500–800 words). Choose one of the following prompts:

(a) Shor's algorithm, when run on a fault-tolerant quantum computer, will break RSA encryption, which secures most of the internet's communication. Estimate the timeline for this threat, describe the countermeasures being developed (post-quantum cryptography), and argue for or against the claim that "quantum computing poses an existential threat to internet security."

(b) Quantum sensing could enable surveillance capabilities (seeing through walls, detecting submarines, monitoring underground facilities) that fundamentally alter the balance of power between states and between states and individuals. Discuss the ethical implications of quantum-enhanced surveillance, drawing parallels to the debate over encryption and privacy.

(c) The quantum computing industry has attracted over $30 billion in investment since 2015, much of it based on projections of future utility that have not yet materialized. Is the quantum technology sector in a "bubble"? Compare to the early internet (1990s) and AI (2010s) investment cycles, and argue for a nuanced position.

Part D: Research Frontier Questions (⭐⭐⭐⭐)

These open-ended questions have no single correct answer and are designed to develop research thinking.

D.1: Your Quantum Roadmap

Choose a specific application of quantum technology (e.g., nitrogen fixation catalyst design using quantum simulation, entanglement-enhanced gravitational wave detection, quantum-secure financial networks). Write a 1–2 page "technology roadmap" that identifies: - The current state of the art. - The key bottlenecks (scientific, engineering, economic). - The milestones that must be achieved. - A realistic timeline with clearly stated assumptions. - The competing classical approaches and why quantum is (or is not) expected to win.

D.2: Literature Deep Dive

Find and read one of the following papers (available on arXiv): - Preskill, "Quantum Computing in the NISQ Era and Beyond" (arXiv:1801.00862) - Degen, Reinhard, and Cappellaro, "Quantum Sensing" (arXiv:1611.02427) - Wehner, Elkouss, and Hanson, "Quantum internet: A vision for the road ahead" (arXiv:1808.03607) - Georgescu, Ashhab, and Nori, "Quantum Simulation" (arXiv:1308.6253)

Write a 2-page critical summary following the paper-reading protocol in Section 30.7. Include: (1) the main question addressed, (2) the key results, (3) the strengths of the paper, (4) the weaknesses or limitations, and (5) one follow-up question or research direction the paper suggests.

D.3: Debate Preparation

Prepare for a structured debate on one of the following propositions: - "Quantum computing will never achieve commercially relevant advantage over classical computing." - "The many-worlds interpretation of quantum mechanics should be adopted as the consensus interpretation." - "Investment in quantum gravity research is a better long-term bet than investment in quantum computing."

For your assigned side (your instructor will assign pro or con), prepare a 5-minute opening statement and anticipate the three strongest counterarguments from the other side. Ground your arguments in the physics discussed in this chapter and the prerequisite chapters (24–29).

D.4: Quantum Ethics Framework

As quantum technologies mature, they raise novel ethical questions (dual-use concerns with quantum computing, surveillance implications of quantum sensing, access inequality in quantum education and workforce). Drawing on Section 30.8 and your own research, draft a one-page "Quantum Technology Ethics Framework" suitable for adoption by a quantum technology company. It should address: (1) responsible disclosure of capabilities and limitations, (2) dual-use risk assessment, (3) equitable access, and (4) environmental impact (cooling requirements, energy consumption).