Fluxonium qubits are superconducting qubits that add a large inductance, usually implemented as an array of Josephson junctions, in parallel with a single small junction and a capacitor. This extra inductive shunt suppresses charge noise sensitivity even further than the transmon and, more importantly, gives the qubit a much larger anharmonicity. The tradeoff is a more complex fabrication process, since the inductive shunt typically needs tens to hundreds of junctions in series.

The circuit is threaded by an external flux, which appears directly in the Hamiltonian as an offset on the phase variable. This makes fluxonium a flux-tunable qubit by construction, unlike the transmon, which is usually frequency-fixed unless a SQUID loop is added separately.

$$H = 4 E_C n^2 - E_J \cos\varphi + \frac{1}{2} E_L (\varphi - \varphi_{ext})^2$$

$E_C$ is the charging energy of the small junction, $E_J$ its Josephson energy, and $E_L$ the inductive energy of the shunt. $\varphi_{ext} = 2\pi \Phi_{ext} / \Phi_0$ is the external flux threading the loop in units of the flux quantum $\Phi_0$. At the “sweet spot” $\varphi_{ext} = \pi$, the double-well potential this Hamiltonian produces gives two nearly degenerate low-energy states separated from higher levels by a large gap, which is what gives fluxonium its long coherence times relative to the transmon.

Fluxonium qubits typically show coherence times an order of magnitude longer than transmons, largely because the large anharmonicity suppresses unwanted transitions to higher levels during gates. The qubit frequency at the sweet spot can also be pushed very low, into the hundreds of MHz, which relaxes some timing constraints on control electronics.

• Advantage: larger anharmonicity, longer coherence times
• Advantage: flux tunability built into the base design
• Disadvantage: more complex fabrication (junction arrays)
• Disadvantage: flux noise sensitivity away from the sweet spot