Anharmonicity

Anharmonicity (or nonlinearity) is the difference between transition frequencies: $\alpha = \omega_{12} - \omega_{01}$ (in energy units). For a transmon, anharmonicity is negative ($\alpha < 0$): $f_{12} < f_{01}$ due to the cosine potential which gets flatter near $\phi = \pi$.

Anharmonicity enables selective qubit control: you can address the $0 \to 1$ transition without driving the $1 \to 2$ transition (if your pulse bandwidth is small enough).

from scqubits import Transmon
 
transmon = Transmon(EJ=15.0, EC=0.3, ncut=30)
anh = transmon.anharmonicity()
print(f"Anharmonicity: {anh:.4f} GHz")
 
# Larger EC/EJ increases anharmonicity
transmon2 = Transmon(EJ=10.0, EC=1.0, ncut=30)
anh2 = transmon2.anharmonicity()
print(f"Larger EC/EJ: {anh2:.4f} GHz")

Strong anharmonicity (large $|\alpha|$):

• Enables fast, selective gates
• More leakage errors (easier to excite to higher levels)
• Harder to suppress (fewer knobs for control)

Weak anharmonicity (small $|\alpha|$):

• Slower gate times (need narrower pulses to avoid 1→2)
• Fewer leakage errors
• Easier to suppress via higher-order corrections

Typical transmons: $\alpha \sim -100$ MHz to $-300$ MHz. Fluxoniums can have stronger anharmonicity ($|\alpha| > 1$ GHz).