Exercises: Chapter 29 — Quantum Computing Systems: Cryogenics, Control Electronics, Calibration, and What It Takes to Run a Quantum Computer

Exercise 29.1: Thermal Population

A transmon qubit with $\omega_q/2\pi = 5.2\ \text{GHz}$ is operated at a mixing chamber temperature of 20 mK. Calculate the equilibrium thermal population of the $|1\rangle$ state. If the temperature rises to 50 mK due to a pulse tube malfunction, what is the new thermal population? At what temperature does $P_{|1\rangle}^{\text{thermal}}$ reach 1%? Derive the general formula and evaluate it for $T = 15\ \text{mK}$, $50\ \text{mK}$, $100\ \text{mK}$, and $300\ \text{mK}$.

Exercise 29.2: Attenuation Budget

Design the attenuation budget for a qubit drive line operating at 5 GHz. The room-temperature source has a noise temperature of 300 K. You need the noise photon number at the qubit to be below $10^{-6}$. Available attenuators provide 3, 6, 10, and 20 dB. The temperature stages are at 50 K, 4 K, 0.7 K, 0.1 K, and 15 mK. Distribute the attenuation across stages to minimize the total while meeting the noise requirement. Justify your choices and calculate the total noise photon number at the qubit.

Exercise 29.3: Ramsey Fit Uncertainty

In a Ramsey experiment with $N = 100$ delay points and 1024 shots per point, the fitted $T_2^*$ is $50 \pm 2\ \mu\text{s}$. How many additional shots would be needed to reduce the uncertainty to $\pm 0.5\ \mu\text{s}$? If the qubit frequency drifts by 10 kHz during the measurement, how does this affect the fitted $T_2^*$? Propose a measurement sequence that is robust to linear frequency drift.

Exercise 29.4: Readout Discrimination

A dispersive readout produces IQ blobs for $|0\rangle$ and $|1\rangle$ with means $\mu_0 = (-1, 0)$ and $\mu_1 = (1, 0)$ (in arbitrary units) and equal covariance matrices $\Sigma = \sigma^2 I$ with $\sigma = 0.5$. Calculate the signal-to-noise ratio SNR = $|\mu_1 - \mu_0|/\sigma$. What is the theoretical assignment infidelity assuming optimal thresholding? If the means shift to $\mu_0 = (-0.8, 0.2)$ and $\mu_1 = (0.9, -0.1)$ due to drift, how much does the infidelity increase if the original threshold is used?

Exercise 29.5: Full System Design

You are the chief engineer for a new 1000-qubit superconducting quantum computer. Estimate: (a) The number of dilution refrigerators needed if each can host 200 qubits with full wiring. (b) The total electrical power consumption (DR + control electronics + cooling). (c) The floor space required. (d) The number of AWG channels and the data rate (in TB/s) between the control system and the qubits, assuming 1 GSa/s, 16-bit samples, and 200 ns gate duration with 50% duty cycle. (e) The annual operating cost, assuming electricity at \$0.10/kWh and one full-time engineer per 100 qubits at \$200K/year fully loaded.

Exercise 29.6: Latency Budget for Error Correction

A surface code error correction cycle requires: (1) measure all data and ancilla qubits, (2) decode the syndrome, (3) apply corrections. If each measurement takes 1 $\mu\text{s}$, the classical decoder takes 3 $\mu\text{s}$, and the correction gates take 0.5 $\mu\text{s}$, what is the total cycle time? If $T_1 = 100\ \mu\text{s}$, what fraction of qubits decay during one cycle? How does this compare to the error threshold for the surface code ($\sim$1%)?

Exercise 29.7: Cooling Power Budget

A dilution refrigerator has a cooling power of $Q = 84\dot{n}_3 T^2$ at the mixing chamber, with $\dot{n}_3 = 10^{-3}\ \text{mol/s}$. The heat loads on the mixing chamber are: 400 coaxial lines at 0.1 $\mu\text{W}$ each, 4 TWPAs at 5 $\mu\text{W}$ each (including pump power), and 100 mW of radiation from the 100 mK stage (attenuated by 60 dB). Calculate the total heat load and determine whether the DR can maintain $T < 20\ \text{mK}$.

Exercise 29.8: Mixer Calibration

An IQ mixer has an amplitude imbalance of $\Delta A = 0.01$ (1%) and a phase skew of $\Delta\phi = 2°$. Calculate the image rejection ratio in dB. If the LO frequency is 5 GHz and the qubit frequency is 5.2 GHz, what is the frequency of the image signal? How much power does the image signal deliver to the qubit if the desired signal power is -100 dBm?

Exercise 29.9: DRAG Pulse Design

A transmon qubit has frequency $\omega_q/2\pi = 5.0\ \text{GHz}$ and anharmonicity $\alpha/2\pi = -300\ \text{MHz}$. Design a DRAG pulse for an $X_\pi$ gate with Gaussian envelope ($\sigma = 10\ \text{ns}$, total duration $4\sigma = 40\ \text{ns}$). Calculate the DRAG correction coefficient $\lambda = -\alpha/(\alpha + \delta)$ for a drive detuning $\delta = 0$. What is the predicted leakage to $|2\rangle$ with and without DRAG?

Exercise 29.10: Crosstalk Mitigation

Two qubits separated by 0.5 mm on a chip have a measured ZZ crosstalk of $\zeta_{ZZ}/2\pi = 50\ \text{kHz}$. If qubit 1 is driven with a $\pi$-pulse of duration 20 ns, what is the phase error on qubit 2? If the crosstalk is cancelled by driving a compensating tone on qubit 2, what amplitude (relative to the qubit 1 drive) is needed? Design a pulse sequence that cancels the ZZ crosstalk to first order.