Fluxonium

Description

The fluxonium qubit, introduced by Manucharyan et al. (2009), consists of a small Josephson junction shunted by a superinductance — a very large inductance ($L\sim 1\mu \text{H}$, corresponding to $E_L/h\sim 0.5-1\text{GHz}$) realized as an array of larger Josephson junctions. This superinductance provides a DC path for phase slips, grounding the superconducting phase and enabling operation at the half-flux-quantum sweet spot ${\Phi }_{\text{ext}}={\Phi }_0/2$, where the qubit transition is first-order insensitive to flux noise.

Unlike the transmon (which operates in the weakly anharmonic oscillator regime), the fluxonium can have enormous anharmonicity — the $|0⟩\to |1⟩$ transition can be as low as $100-1000\text{MHz}$ while higher transitions are at $5-8\text{GHz}$. This spectral isolation makes leakage errors fundamentally smaller. The tradeoff is that the low transition frequency makes direct dispersive readout more challenging, typically requiring auxiliary readout schemes or coupling to a transmon for measurement.

In the “heavy fluxonium” regime ($E_C/h\sim 0.5-1\text{GHz}$, $E_J/E_C\sim 3-8$), the wavefunctions of $|0⟩$ and $|1⟩$ have disjoint support in phase space: $|0⟩$ is localized in one well of the cosine potential and $|1⟩$ in the other. This disjoint support makes the qubit insensitive to virtually all local noise operators, enabling $T_1$ times exceeding $1\text{ms}$.

Figure

Hamiltonian

$H=4E_C{\hat{n}}^2-E_J\cos \hat{\phi }+\frac{1}{2}{E}_L(\hat{\phi }-2\pi {\Phi }_{\text{ext}}/{\Phi }_0{)}^2$

where $E_C=e^2/2C$ is the charging energy, $E_J$ is the Josephson energy of the small junction, $E_L=({\Phi }_0/2\pi {)}^2/L$ is the inductive energy of the superinductance, $\hat{n}$ and $\hat{\phi }$ are conjugate charge and phase operators, and ${\Phi }_{\text{ext}}$ is the external flux threading the loop.

At the half-flux-quantum sweet spot (${\Phi }_{\text{ext}}={\Phi }_0/2$), the potential has a double-well structure with the two minima related by parity symmetry $\phi \to -\phi +2\pi$. The qubit states are the symmetric and antisymmetric superpositions of the states localized in each well, split by the tunneling rate through the cosine barrier.

Motivation

The transmon’s achilles heel is its weak anharmonicity ($\sim -200\text{MHz}$), which limits gate speeds and makes it vulnerable to leakage to $|2⟩$. The fluxonium addresses this with GHz-scale anharmonicity while simultaneously achieving superior coherence through flux-sweet-spot operation and disjoint-support noise protection. The main challenge is implementing fast, high-fidelity gates at low qubit frequencies and achieving high-fidelity readout without a direct dispersive shift.

Key Findings

• Superinductance via Josephson junction arrays provides $L>100\text{nH}$ with self-resonance above qubit operating frequencies.
• At ${\Phi }_{\text{ext}}={\Phi }_0/2$, first-order flux noise sensitivity vanishes (sweet spot).
• Heavy fluxonium regime achieves $T_1>1\text{ms}$ through disjoint-support protection.
• Two-qubit gates demonstrated via direct capacitive coupling and inductive coupling schemes.
• Recent results show single-qubit gate fidelities $>99.99\%$ and two-qubit gate fidelities $>99.7\%$.

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_1$	100 μs – 1.5 ms	Heavy fluxonium at half-flux sweet spot	Manucharyan 2009
$T_2$ (echo)	100–500 μs	Echo at sweet spot	Somoroff et al. 2023
Anharmonicity	3–8 GHz	${\omega }_{12}-{\omega }_{01}$; much larger than transmon	Manucharyan 2009
Qubit frequency ${\omega }_{01}/2\pi$	100 MHz – 1 GHz	Much lower than transmon	—
$E_J/E_C$	3–8	Heavy fluxonium regime	—
1Q gate fidelity	99.97–99.998%	Microwave or charge-parity-protected gates; MIT 2025 SOTA 99.998%	Ding et al. 2023, Somoroff et al. 2023
2Q gate fidelity	99.2–99.92%	Capacitive or inductive coupling; CZ 99.92%	Ding et al. 2023
Gate time (1Q)	20–100 ns	Frequency-dependent	—
Operating temperature	10–20 mK	Dilution refrigerator	—

Linked Papers

• manucharyan-2009-fluxonium

Related Entries

• transmon
• flux-qubit
• cooper-pair-box-charge-qubit
• circuit-qed
• blochnium