Fluxonium is a superconducting qubit with stronger nonlinearity than transmons. It combines a Josephson junction with an L-C oscillator in series. Fluxoniums are flux-tunable (controlled by applied magnetic flux), compact, and exhibit strong anharmonicity—useful for fast, robust gates.

Fluxonium Hamiltonian includes kinetic energy (from the series inductor), potential energy (from Josephson junction), and a flux-dependent term:

$$H = 4E_C \hat{n}^2 + E_L (\hat{\phi} - \Phi_{\text{ext}}/\Phi_0)^2 - E_J \cos(\hat{\phi})$$

where $E_L$ is the loop inductance, $\Phi_{\text{ext}}$ is applied flux, $\Phi_0$ is flux quantum.

from scqubits import Fluxonium
 
fluxonium = Fluxonium(
    EJ=12.0,            # Josephson energy (GHz)
    EC=2.5,             # Charging energy (GHz)
    EL=0.5,             # Loop inductance energy (GHz)
    flux=0.5,           # External flux in flux quanta (0 to 1)
    ncut=30,
    flux_cut=30
)
 
# Sweep flux
fluxes = np.linspace(0, 1, 100)
freq_vs_flux = []
for phi in fluxes:
    fluxonium.flux = phi
    freq_vs_flux.append(fluxonium.f_01())
 
print(freq_vs_flux)

Fluxoniums show strong flux-dependence of frequency—useful for tunable couplings and parametric drives. Their strong anharmonicity enables high-speed gates with reduced leakage errors.

Fluxoniums are gaining popularity in quantum computing platforms (Rigetti) and quantum simulation.