In a transmon, the frequency of the |0⟩→|1⟩ transition is 5.0 GHz and the anharmonicity (the |1⟩→|2⟩ frequency minus the |0⟩→|1⟩ frequency) is −200 MHz. What is the frequency of the |1⟩→|2⟩ transition?

Answer: A. 4.8 GHz

The |1⟩→|2⟩ frequency is 5.0 GHz + (−0.2 GHz) = 4.8 GHz. Because of the nonlinearity of the Josephson junction, the level spacing gets slightly smaller for higher levels, so the |1⟩→|2⟩ frequency is lower. A 5.0 GHz microwave drive is detuned from the 4.8 GHz transition, so the qubit is less likely to leak into |2⟩, outside the qubit subspace. However, if the pulses are made too short, their frequency spread approaches 200 MHz and leakage to |2⟩ increases, so the pulse shape is carefully designed.

• A. Correct. 5.0 GHz − 0.2 GHz = 4.8 GHz.
• B. This is the case of a linear LC resonator with equally spaced levels, where the |1⟩→|2⟩ transition would also be driven at the same frequency.
• C. This reverses the sign of the anharmonicity. A transmon's anharmonicity is negative, so higher transitions have lower frequencies.