Exercises: Chapter 26 — Superconducting Qubits: Transmons, Flux Qubits, and the Hardware Inside IBM and Google Quantum Computers

Exercise 26.1: Transmon Energy Levels. Calculate the first four energy levels of a transmon with $E_J/h = 20$ GHz and $E_C/h = 0.25$ GHz. Compute the anharmonicity $\alpha = E_{12} - E_{01}$ and the relative anharmonicity $\alpha_r = \alpha / E_{01}$. At what Rabi frequency would leakage to $|2\rangle$ exceed 1% for a resonant $\pi$-pulse? (Hint: Use the Duffing oscillator approximation and the DRAG condition.)

Exercise 26.2: Dispersive Readout SNR. A transmon qubit ($\omega_q/2\pi = 5.0$ GHz) is coupled to a readout resonator ($\omega_r/2\pi = 7.0$ GHz) with coupling $g/2\pi = 100$ MHz. The resonator linewidth is $\kappa/2\pi = 5$ MHz. Calculate the dispersive shift $\chi$. If the measurement uses $\bar{n} = 5$ photons and integrates for $\tau = 400$ ns, what is the expected SNR? What integration time is needed for SNR = 10?

Exercise 26.3: Cross-Resonance Gate. For two fixed-frequency transmons with frequencies $\omega_1/2\pi = 5.0$ GHz and $\omega_2/2\pi = 5.3$ GHz, coupled with $g/2\pi = 15$ MHz, compute the $ZZ$ crosstalk strength $\zeta_{ZZ}$ assuming $\alpha/2\pi = -300$ MHz for both qubits. If the CR gate requires a $ZX$ rotation of $\pi/2$ and the drive-induced $ZX$ rate is $\Omega_{ZX}/2\pi = 2$ MHz, what is the gate duration? What is the accumulated $ZZ$ phase error during the gate?

Exercise 26.4: Frequency Crowding Optimization. You have 10 fixed-frequency transmons to place in the band 4.5–5.5 GHz. Each qubit needs a minimum spacing of 50 MHz from every other qubit, and the readout resonators (one per qubit) need 100 MHz spacing from each other and from all qubit frequencies. Propose a frequency plan. How many qubits can you fit if the minimum spacing is reduced to 30 MHz?

Exercise 26.5: Purcell Filter Design. A qubit with $T_1 = 300\ \mu\text{s}$ is limited by Purcell decay through a readout resonator with $\kappa/2\pi = 10$ MHz. The qubit–resonator detuning is $\Delta_{qr}/2\pi = 2$ GHz and coupling is $g/2\pi = 100$ MHz. Calculate the Purcell-limited $T_1$. Design a Purcell bandpass filter (specify its center frequency and bandwidth) that would increase the Purcell limit to above 10 ms.

Exercise 26.6: Charge Dispersion Suppression. For a transmon with $E_J/E_C = 50$, compute the charge dispersion $\varepsilon_1$ using the approximate formula. Compare this to a Cooper pair box with $E_J/E_C = 1$. How many orders of magnitude improvement in charge noise insensitivity does the transmon provide?

Exercise 26.7: Surface Code Threshold Estimate. Using the threshold formula $p_L \approx 0.1 \times (p/p_{\text{th}})^{d/2}$, estimate the logical error rate for a distance-$d$ surface code at physical error rate $p = 10^{-3}$. For what distance $d$ does the logical error rate drop below $10^{-10}$? How many physical qubits are needed per logical qubit?

Exercise 26.8: Readout Fidelity Optimization. Derive the optimal integration time for dispersive readout that maximizes the assignment fidelity given a fixed measurement-induced dephasing rate $\Gamma_\phi^{\text{meas}} = 4\chi^2\bar{n}\kappa / (\kappa^2 + 4\Delta^2)$. Show that the optimal integration time scales as $\tau_{\text{opt}} \propto 1/\sqrt{\bar{n}}$ and that the resulting fidelity scales as $1 - e^{-\text{SNR}}$ where $\text{SNR} \propto \chi\sqrt{\bar{n}\kappa\tau}$.

Exercise 26.9: Tunable Coupler Analysis. For a tunable coupler with frequency $\omega_c/2\pi = 7$ GHz, coupling $g_1/2\pi = g_2/2\pi = 50$ MHz to two qubits at $\omega_1/2\pi = 5$ GHz and $\omega_2/2\pi = 5.2$ GHz, compute the effective qubit-qubit coupling $g_{\text{eff}}$ when the coupler is (a) at 7 GHz, (b) at 6 GHz, and (c) at 5.1 GHz. What happens when the coupler frequency equals one of the qubit frequencies?

Exercise 26.10: Coherence Budget. A transmon has $T_1 = 300\,\mu$s and $T_2^* = 100\,\mu$s. Decompose the total dephasing rate $1/T_2^*$ into contributions from $T_1$ and pure dephasing $T_\phi$: $1/T_2^* = 1/(2T_1) + 1/T_\phi$. Compute $T_\phi$. If $T_1$ improves to 500 $\mu$s (by eliminating dielectric loss) but $T_\phi$ stays the same, what is the new $T_2^*$?