# How do the extra energy levels of a transmon qubit affect computation/fidelity?

I was reading about transmon qubits, and I know that they are not true two-level systems. Are there any math/papers which talk about how those extra energy levels affect the computation? I'm assuming it may have a small effect.

• The IBM Qiskit textbook appears to have some information about how to suppress the higher energy levels to improve the fidelity which may answer your question. In particular, sections 6.2 and 6.3 might be useful to you. May 23 at 1:23

When non-computational states are unintentionally populated, this is called leakage, and it is one of the leading sources of error in transmon qubits. Originally, the transmon emerged from an earlier design called the Cooper-pair box, which suffered from charge noise. By adding a large shunt capacitor, the sensitivity of the device to charge noise was greatly reduced, at the expense of lowering the anharmonicity. The anharmonicity is the difference between the 1-2 energy level spacing to the 0-1 energy level spacing. A perfect two-level system has infinite anharmonicity (the next accessible level is infinitely far away from the computational space, so there is no chance of leakage), and the worst possible "two"-level system has 0 anharmonicity (one example is a harmonic oscillator). Transmon qubit frequencies are usually a couple GHz, while their anharmonicities are a couple hundred MHz (and negative, which means $$|2\rangle$$ is closer to $$|1\rangle$$ than $$|0\rangle$$).