Integer Fluxonium Qubit

• Integer Fluxonium is a superconducting qubit architecture designed at zero external flux bias, utilizing a small Josephson junction, large superinductance, and capacitive shunt.
• It achieves high anharmonicity and robust coherence through symmetry-protected operation and exponentially suppressed tunneling, resulting in prolonged T1 and T2 times.
• The design supports fast, high-fidelity single- and two-qubit gate operations and enhanced readout protocols, making it a promising candidate for scalable quantum processors.

Integer fluxonium refers to a superconducting qubit architecture based on the fluxonium circuit that is designed and operated at zero external magnetic flux bias. Unlike traditional fluxonium designs biased at half a flux quantum, integer fluxonium exploits specific symmetry points of the circuit Hamiltonian at integer multiples of the magnetic flux quantum. The combination of a small Josephson junction, a large superinductance, and capacitive shunt yields a qubit with high anharmonicity, enhanced resilience to dissipation, and partially protected quantum coherence. Integer fluxonium bridges key advantages of transmon-frequency operation, robust coherence, and gate controllability, positioning it as a strong candidate for scalable superconducting quantum information processors.

1. Circuit Architecture and Hamiltonian Formulation

The integer fluxonium qubit is constructed from a superconducting loop shunted by a small Josephson junction (energy $E_J$), a large inductance (superinductance, $L$), and a capacitance $C$. The canonical Hamiltonian takes the form

$$H = 4 E_C \, \hat{n}^2 - E_J \cos(\hat\phi - \phi_{\mathrm{ext}}) + \frac{1}{2} E_L \hat\phi^2$$

where

• $E_C = e^2/(2C)$ is the charging energy,
• $E_J$ is the Josephson energy,
• $E_L = (\hbar/2e)^2/L$ is the inductive energy,
• $\hat\phi$ is the superconducting phase across the junction, and
• $\hat{n}$ is the conjugate Cooper-pair number operator, with $[\hat\phi, \hat{n}] = i$.

For integer fluxonium, the external flux bias is set to $\phi_{\mathrm{ext}} = 0$ (i.e., zero magnetic field). In the regime $E_L \ll E_J$, the resulting potential

$U(\phi) = -E_J \cos(\phi) + \frac{1}{2} E_L \phi^2$

exhibits multiple local minima at integer multiples of $2\pi$, forming energetically well-separated persistent-current states. The low-energy dynamics are governed by quantum tunneling between these minima, with tunneling amplitudes (for single- and double-phase slips) that are exponentially suppressed for increasing $E_J/E_C$.

An approximate tight-binding form for the effective Hamiltonian in the basis of fluxon number states $|m\rangle$ is

$$H_{\mathrm{eff}} = \sum_m \frac{1}{2} E_L^\Sigma (2\pi m - \phi_{\mathrm{ext}})^2 |m\rangle\langle m| - \frac{\epsilon_1}{2} \sum_m (|m\rangle\langle m{+}1| + \text{h.c.}) + \frac{\epsilon_2}{2} \sum_m (|m\rangle\langle m{+}2| + \text{h.c.})$$

with $E_L^\Sigma = (E_L^{-1} + E_J^{-1})^{-1}$, and tunneling amplitudes $\epsilon_1$, $\epsilon_2$ characterizing the strengths of single and double fluxon tunneling, respectively (Mencia et al., 25 Mar 2024, Ardati et al., 7 Feb 2024).

2. Coherence Mechanisms and Error Suppression

Integer fluxonium qubits demonstrate strong coherence due to a combination of circuit symmetry and exponentially suppressed matrix elements for energy relaxation. Critically, at the integer sweet spot:

• The transition matrix element $\langle 0|\hat{n}|1\rangle$ between ground and first excited states is small, significantly reducing sensitivity to dielectric loss even for materials with modest loss tangents (quality factors in the $10^5$ range).
• The transition from $|0\rangle$ to $|2\rangle$ is parity-forbidden at $\phi_{\mathrm{ext}} = 0$, further protecting the computational manifold from leakage (Mencia et al., 25 Mar 2024).

Measured relaxation times $T_1$ exceeding 100–170 $\mu$s and Ramsey/echo dephasing times $T_2^* > 100\,\mu$s have been reported, despite underlying materials limitations. The decay rates obey

$$\Gamma_{jk}^{(\mathrm{diel})} = \frac{8 E_C}{\hbar} |\langle j | \hat{n} | k \rangle|^2 \tan{\delta_C} \left[ 1 + \coth\left(\frac{\hbar \omega_{jk}}{2k_B T}\right) \right]$$

where $\tan{\delta_C}$ is the dielectric loss tangent (Somoroff et al., 2023). Because the relevant charge matrix element is reduced by factors of $10$–$20$ compared to transmons, integer fluxonium’s coherence outperforms expectations based on component quality factor alone (Mencia et al., 25 Mar 2024).

Additionally, the disjoint wavefunction support of fluxon-number basis states yields minimal overlap between $|0\rangle$ and $|1\rangle$—a feature rooted in bifluxon (double phase-slip) tunneling processes at the sweet spot, which ensure relaxation channels are highly suppressed (Ardati et al., 7 Feb 2024). Phase-flip errors (dephasing) remain dictated by environmental flux and photon shot noise, yet are not exacerbated by this separation.

3. Gate Control and Noise-Protected Operations

Integer fluxonium supports fast, high-fidelity single-qubit gates using conventional microwave pulses as well as sub-harmonic parametric driving. Microwave-driven Clifford gates with average fidelities above $99.9\%$ were demonstrated with pulse durations as short as $\sim$88 ns (Mencia et al., 25 Mar 2024). Notably, the anharmonicity of the spectrum ($\omega_{12} \gg \omega_{01}$) allows these gates to be realized without the leakage suppression techniques (e.g., DRAG) required for weakly anharmonic qubits.

For enhanced noise protection, sub-harmonic parametric drives—where the qubit is coherently manipulated via n-photon transitions at $\omega_{eg}/n$ ($n\geq 3$), mediated by flux-line control—allow all fast control to be routed through a protected low-pass filtered flux interface (Schirk et al., 1 Oct 2024). This architecture:

• Suppresses the environmental density of states at the qubit frequency by filtering, protecting $T_1$ from control-line-induced decay.
• Enables coherent Rabi oscillations and full single-qubit control (even up to 11-photon resonant transitions), with gate fidelities exceeding $99.94\%$ benchmarked for 3-photon subharmonic drives.
• Achieves $T_1$ improvements by a factor of five and $T_2^{\mathrm{echo}}$ improvements of a factor of ten compared to unfiltered setups, illustrating that protection does not compromise fast gating.

These advances demonstrate that strong circuit nonlinearity in fluxonium supports both robust gate operations and effective protection from control-line dissipation.

4. Two-Qubit Coupling, Scalability, and Gate Schemes

Integer fluxonium qubits are compatible with a variety of high-fidelity two-qubit gate schemes tailored to their spectrum, partially protected nature, and compatibility with scalable circuit layouts.

• Transmon-assisted fluxonium–transmon–fluxonium (FTF) architectures have been proposed and analyzed, enabling two integer fluxonium qubits to interact via a transmon coupler (Wang et al., 5 Sep 2025).
• Two key CZ gate schemes have been demonstrated:
  • Flux-activated adiabatic CZ: Applying a dynamic flux pulse to the coupler, the system transiently activates a strong ZZ interaction so that the conditional phase

  $\theta_{ZZ} = \int_0^{T_g} \zeta(t) dt = (1+2N)\pi$

  accumulates during the gate. Optimized pulse shaping ensures coherent errors below $10^{-6}$ within tens of nanoseconds. - Microwave-activated non-adiabatic geometric CZ: A selective microwave drives the coupler near a specific resonance to perform a $2\pi$ Rabi rotation between chosen states, imparting the needed conditional phase. The protocol achieves error rates in the $10^{-6}$ regime, with leakage suppressed by the large energy gap to higher states.

• The FTF architecture can be tuned such that the residual static ZZ interaction is canceled via destructive interference between direct and coupler-mediated contributions, thus eliminating always-on entanglement during idle periods (Wang et al., 5 Sep 2025).

These features allow integration of integer fluxonium with scalable arrangements, low crosstalk, and support for surface-code-compatible gate primitives.

5. Readout Enhancement and Measurement Protocols

The combination of strong protection at the sweet spot and large anharmonicity poses unique challenges for fast and high-fidelity readout, particularly since standard dispersive readout schemes at low qubit frequencies yield slow integration times and may be thermally limited. Dynamically pulse-assisted readout, whereby a fast flux pulse is applied during the measurement window to transiently enhance the qubit–resonator coupling (dispersive shift $\chi$), increases the readout contrast by $\sim$20%, enabling:

• Readout assignment fidelities of 94.3% within 280 ns integration time (without quantum-limited amplifiers) and SNR-limited fidelities up to 99.9% in 360 ns (Stefanski et al., 20 Nov 2024).
• Performance that meets or exceeds previous fluxonium readout benchmarks, including those that required parametric amplifiers.
• Additional prospects for higher measurement efficiency via integration of quantum-limited amplifiers and further pulse sequence optimization.

This approach leverages the flux-tunability intrinsic to fluxonium for improved measurement, addressing a critical bottleneck for practical algorithms requiring fast and mid-circuit readout.

6. Noise Bias, Protected Encoding, and Passive Memory

Integer fluxonium qubits—especially in the heavy-fluxonium regime—naturally realize a $\mathbb{Z}_2$-symmetry-broken ground-state sector comprising two persistent-current macroscopic quantum states. Variational analysis in the oscillator (Fock) basis establishes that the logical basis states (macroscopically separated persistent current states) can be represented as squeezed coherent states with displacement

$$\alpha \simeq \frac{\pi}{2}\left(\frac{E_L}{2E_C}\right)^{1/4}\frac{E_J}{E_J + E_L}$$

and optimal squeezing parameter

$\theta \simeq \frac{1}{4} \ln \frac{E_J}{E_L}$

(Lieu et al., 27 Jan 2025). The overlap between these logical states is exponentially small in $E_J/E_C$, resulting in

• Bit-flip error rates that are exponentially suppressed: $$\mathrm{Rate}_{\mathrm{bitflip}} \sim \exp(-4|\alpha'|^2)$$,
• Phase-flip rates (from dephasing noise) that do not increase with $\alpha$.

This pronounced error bias is analogous to that of $\cos(2\theta)$ and cat-qubit systems. Coupling multiple protected fluxonium qubits in a 2D geometry with engineered Ising-like interactions holds potential for realizing passive, Hamiltonian-protected quantum memories that can suppress both bit-flip and phase-flip errors exponentially (Lieu et al., 27 Jan 2025).

7. Implications for Scalable Quantum Computing

Integer fluxonium qubits have demonstrated that a single-mode superconducting qubit, when engineered at the symmetry-protected sweet spot, can achieve:

• High-fidelity single- and two-qubit gates with error rates at or below $10^{-4}$ ($\mathcal{F}_1 > 0.999$, $\mathcal{F}_{\mathrm{CZ}} \sim 0.999999$),
• Robust coherence exceeding 100 μs in $T_1$ and $T_2^*$ without stringent material requirements,
• Circuit-level error suppression mechanisms that are rooted in wavefunction parity and controlled fluxon tunneling,
• Compatibility with microwave circuit QED, parametric gate control, and large-scale planar integration (Mencia et al., 25 Mar 2024, Wang et al., 5 Sep 2025, Nguyen et al., 2022).

Integration of integer fluxonium into transmon-assisted gate schemes, noise-protected control, and rapid readout protocols define a promising path toward large-scale quantum processors that feature partially protected hardware, addressable error bias, and feasible fabrication and control requirements. Continued advances in fast flux control, coupler designs, and error-corrected architectures are anticipated to further lower logical error rates and facilitate robust quantum computation using integer fluxonium as a foundational qubit element.