Exercises: Chapter 18 — The NISQ Era — Noisy Intermediate-Scale Quantum Computers and the Algorithms Designed for Imperfect Hardware

Exercise 18.1: Noise Model Exploration

Modify the noise model in Section 18.9 to explore the effect of varying two-qubit gate error rates. Plot the Bell state fidelity as a function of $p_2$ (the two-qubit depolarizing probability) for $p_2 \in [0, 0.1]$. At what error rate does the fidelity drop below 0.5? Below 0.1?

Exercise 18.2: Quantum Volume Estimation

A device has 20 qubits with single-qubit gate fidelity 99.95%, two-qubit gate fidelity 99.5%, and measurement fidelity 99.0%. Estimate the maximum circuit depth for which the success probability exceeds 1/3. Based on this, estimate the quantum volume. (Assume a square circuit structure with one two-qubit gate per qubit per layer.)

Exercise 18.3: Circuit Knitting Overhead

A 10-qubit circuit needs to be run on a 5-qubit device. If the circuit is cut into two 5-qubit fragments with 3 wire cuts, and each cut requires a quasi-probability decomposition with $L_1$-norm overhead $c = 6$, how many total circuit executions are needed to achieve the same statistical precision as a single uncut execution? Discuss the practical implications.

Exercise 18.4: Error Mitigation by Averaging

Consider a simple error mitigation strategy: run the same circuit $N$ times and average the results. Show that the variance of the estimated expectation value scales as $1/N$. Why is this insufficient for deep circuits where the signal is exponentially suppressed?

Exercise 18.5: NISQ vs. Fault-Tolerant Trade-offs

Estimate the number of physical qubits needed to run a 100-qubit, depth-1000 circuit (a) on a NISQ device with no error correction, and (b) on a fault-tolerant device using the surface code with physical error rate $10^{-3}$ and threshold $10^{-2}$. Assume 1,000 physical qubits per logical qubit for the surface code. Discuss the trade-offs in terms of qubit count, operation speed, and algorithmic flexibility.

Exercise 18.6: Barren Plateau Analysis

Consider a hardware-efficient ansatz with $L$ layers of single-qubit rotations and entangling gates on $n$ qubits. Show that the gradient variance for a global cost function $C(\boldsymbol{\theta}) = \langle Z^{\otimes n} \rangle$ scales as $O(1/2^n)$ for $L = O(\text{poly}(n))$. What happens if the cost function is local, e.g., $C(\boldsymbol{\theta}) = \frac{1}{n} \sum_i \langle Z_i \rangle$?

Exercise 18.7: ZNE Implementation

Implement zero-noise extrapolation for the expectation value $\langle XX + ZZ \rangle$ on a Bell state. Use unitary folding with scale factors 1, 3, 5 and perform (a) linear extrapolation, (b) quadratic extrapolation, and (c) Richardson extrapolation. Compare the three methods. How close does each get to the ideal value of 2?

Exercise 18.8: Readout Error Mitigation

A 2-qubit device has readout error matrix $M$ where each qubit independently flips with probability $p = 0.03$. Compute $M^{-1}$ and apply readout error mitigation to the noisy Bell state measurements from Section 18.9. How much does the corrected $\langle ZZ \rangle$ improve?

Exercise 18.9: Coherence Budget

A superconducting qubit has $T_1 = 150$ µs, $T_2 = 100$ µs, single-qubit gate time $t_1 = 30$ ns, and two-qubit gate time $t_2 = 60$ ns. Estimate the maximum number of (a) single-qubit gates and (b) two-qubit gates that can be executed before decoherence reduces the state fidelity to $1/e \approx 37\%$. Assume gate errors are negligible compared to decoherence.

Exercise 18.10: Quantum Volume Calculation

Design a quantum volume experiment for a 4-qubit device. Generate 5 random SU(4) circuits of depth 4, compute their heavy output sets (using a noiseless simulator), and estimate what fraction of heavy outputs a noisy simulator would produce. How does this compare to the 2/3 threshold?