A qubit with a frequency of 5 GHz (hf/k_B ≈ 0.24 K) is in thermal equilibrium at an effective temperature of 50 mK. Which is closest to the ratio of the probabilities of being in |1⟩ and in |0⟩, exp(−hf/k_B T)?

Answer: B. About 0.8%

hf/k_B T = 0.24 K ÷ 0.05 K = 4.8, so the ratio is exp(−4.8) ≈ 0.0082, about 0.8%. Even when the refrigerator is at about 10 mK, noise entering through the wiring and other effects can make the effective temperature of the qubit higher, and the fraction left in |1⟩ before a computation starts becomes an initialization error. For this reason, active initialization, which measures the state and then returns the qubit to |0⟩, is sometimes used. At 20 mK the ratio drops to about 6×10⁻⁶, showing that temperature has an exponential effect.

• A. This is the value at 20 mK (exp(−12) ≈ 6×10⁻⁶).
• B. Correct. exp(−4.8) ≈ 0.0082.
• C. This uses k_B T/hf ≈ 0.21 directly and misses the exponential form.