Inductively Shunted Transmons

• Inductively shunted transmons are superconducting circuits that combine a Josephson junction with a parallel inductor to provide parabolic confinement and robust stability under strong drive.
• They offer tunable energy spectra and anharmonicity through controlled magnetic flux and optimized circuit parameters, suppressing parasitic interactions and noise.
• The design supports scalable quantum architectures by achieving high gate fidelities, enabling QND readout and passive suppression of unwanted couplings.

An inductively shunted transmon is a superconducting artificial atom in which a Josephson junction is embedded in a circuit with both capacitive and inductive (usually geometric or kinetic-inductance) shunts. This architecture extends the conventional transmon design by placing a linear inductance in parallel with the Josephson junction and shunt capacitance. The inductive shunt fundamentally alters the confining potential, leading to enhanced robustness to strong parametric driving, improved noise resilience, tunable nonlinearity, and new regimes of circuit quantum electrodynamics (cQED).

1. Circuit Topology and Hamiltonian Structure

The canonical inductively shunted transmon consists of a single Josephson junction with energy $E_J$ and capacitance $C$, shunted by a geometric or kinetic inductor $L$. An external magnetic flux $\Phi_\mathrm{ext}$ threads the loop. The Hamiltonian in the charge-phase basis reads: $$\hat{H} = 4E_C \hat{n}^2 - E_J\cos(\hat{\varphi}+\varphi_e) + \frac{1}{2}E_L\hat{\varphi}^2,$$ where $E_C = e^2/(2C)$, $E_L = (\Phi_0/2\pi)^2/L$, $\varphi_e = 2\pi\Phi_\mathrm{ext}/\Phi_0$, $[\hat{\varphi}, \hat{n}] = i$ (Fasciati et al., 2024, Kalacheva et al., 2023, Hassani et al., 2022). The circuit may be extended to include additional coupling branches, resonators, or multiple transmons connected via shared inductances (Dumur et al., 2015, Richer et al., 2017).

The key impact of the inductive term is to provide a parabolic confinement for the superconducting phase across the junction, breaking the $2\pi$ periodicity of the Josephson term at high energies. This results in (i) continuous confinement even under strong drive, (ii) modification of the energy levels (anharmonicity), (iii) significant suppression of phase fluctuations at large excitation energies, and (iv) tunable sensitivity to magnetic and charge noise.

2. Energy Spectrum, Anharmonicity, and Flux Dependence

For $E_J/E_C \gg 1$, the system operates in the transmon-like regime, but the addition of $E_L$ tunes the eigenstate structure. The energy levels are determined by numerical diagonalization of $\hat{H}$. In the presence of a flux bias, the spectrum exhibits the following features (Fasciati et al., 2024, Kalacheva et al., 2023, Hassani et al., 2022, Duda et al., 28 Apr 2025):

• The qubit transition ($f_{01}$) varies with $\Phi_\mathrm{ext}$, with typical experimentally observed ranges such as $f_{01} = 6.98\,\mathrm{GHz}$ at $\Phi_\mathrm{ext}=0$ to $3.67\,\mathrm{GHz}$ at $\Phi_\mathrm{ext} = \Phi_0/2$.
• The anharmonicity, $\alpha = E_{12}-E_{01}$, can be tuned over a wide range and can change sign ($\alpha > 0$ or $\alpha < 0$), with values $\alpha_{\mathrm{max}} \sim +228\,\mathrm{MHz}$ at $\Phi_\mathrm{ext} = \Phi_0/2$ and $\alpha \lesssim -200\,\mathrm{MHz}$ at integer flux.
• The flux dispersion of the plasmonic (single-well) spectrum is suppressed quadratically in $E_J/E_L$, yielding "sweet-spot everywhere" behavior with measured dispersions as low as $5.1\,\mathrm{MHz}$ over a full flux quantum (Hassani et al., 2022).
• At certain flux values, the spectrum can become nearly harmonic ($\alpha \approx 0$), enabling new operational modes for multi-photon protocols and interaction suppression.

A succinct table summarizing this tunability:

Parameter	Typical Range	Implications
$E_J/E_C$	10–100	Charge noise suppression
$E_L/E_J$	$\sim$0.5–2	Level of quadratic confinement
$f_{01}$	1–10 GHz (design-dependent)	Qubit operational frequency
$\alpha$	$-\!300$ MHz to $+\!300$ MHz	Gate selectivity; interaction engineering

3. Driven Dynamics, Instability Suppression, and Robustness

A primary motivation for the inductive shunt is the stabilization of Josephson circuits under strong parametric drive. In bare transmons, high drive amplitude "ionizes" the mode, ejecting population into the unconfined phase space: this destroys the qubit's nonlinearity and leads to decoherence, as evidenced by Floquet–Markov analysis and experiment (Verney et al., 2018). For bare transmons, above a critical photon number $n_c$ ($\sim 100–300$), the device undergoes a sudden loss of spectral anharmonicity, and steady-state purity plummets.

The inductive shunt prevents this instability by confining the phase variable with the harmonic term $E_L\hat{\varphi}^2/2$, so that even strong pumping (large AC drive amplitude) cannot delocalize the qubit. Floquet–Markov simulations show that the purity of the steady state remains $>97\%$ for all experimentally accessible pump powers, and the AC Stark shift evolves smoothly without spectral collapse (Verney et al., 2018). The parabolic confinement ensures the system never leaves the nonlinear regime, facilitating advanced Hamiltonian engineering protocols and stable operation with pump-induced Kerr tuning.

4. Coupling Schemes and Gate Operations

Inductively shunted transmons offer enhanced flexibility for two-qubit coupling, readout, and gate operations. Capacitively coupling an IST to a conventional transmon enables full suppression of parasitic $ZZ$ interactions by tuning one device to have positive and the other negative anharmonicity—the cross dispersion $\zeta$ cancels when $\alpha_1 = -\alpha_2$ (Fasciati et al., 2024). This passive suppression of $ZZ$ is confirmed by full diagonalization and experiment, achieving $|\zeta|/2\pi < 5\,\mathrm{kHz}$, below typical decoherence rates.

Away from flux sweet spots, cubic nonlinearity permits first-order sideband transitions, enabling fast entangling gates such as bSWAP ($|00\rangle \leftrightarrow |11\rangle$ in 125 ns), iSWAP, and high-fidelity (interleaved RB $F_\mathrm{CZ} = 95.8\%$) controlled gates (Fasciati et al., 2024). Red sideband gates are implemented via frequency-selective microwave drives, and coherent control is robust to moderate flux noise ($A_\phi \approx 6.8\,\mu\Phi_0$). Design simplicity and parametric tunability yield a broad gate toolbox without extra coupling elements (Richer et al., 2017).

Tunability between pure transverse and pure longitudinal coupling is accomplished using additional Josephson and geometric elements: at specific external fluxes, the coupling can be switched from $\sigma_x$- (transverse) to $\sigma_z$- (longitudinal) type, facilitating both standard cQED operations and QND readout via longitudinal coupling (Richer et al., 2017, Richer, 2018):

Coupling Type	Tuning Condition	Physical Mechanism
Transverse ($\sigma_x$)	$\varphi_x=0$	$g_{xx}\neq0$, $g_{zx}=0$
Longitudinal ($\sigma_z$)	$\varphi_x=k\pi/2$	$g_{zx}\neq0$, $g_{xx}=0$

5. Noise Robustness, Coherence, and Sweet Spot Physics

The addition of a large inductance suppresses both charge and flux noise sensitivity:

• Charge Noise: The large $E_J/E_C$ ratio, typical of both transmon and IST designs, suppresses charge dispersion exponentially for all relevant levels, and the inductive shunt provides further isolation (Kalacheva et al., 2023, Hassani et al., 2022).
• Flux Noise: In the IST regime ($E_J/E_L\gg1$), the leading flux dependence of plasmon transition frequencies is quadratic, not linear, in $\Phi_\mathrm{ext}$, yielding “sweet-spot everywhere” operation. Devices with $E_J/E_L\gtrsim 50$ exhibit flux dispersions of a few MHz over a full flux quantum and near-constant $T_1$ and $T_2^*$ throughout the bias range (Hassani et al., 2022, Duda et al., 28 Apr 2025).
• Measured Coherence: Experimental ISTs and kinemons report $T_1=9$–$20\,\mu$s, $T_2^*=7$–$17\,\mu$s; conventional transmons fabricated on the same platform show $T_1\approx14\,\mu$s, $T_2^*\approx8\,\mu$s, indicating no degradation due to $L$ (Kalacheva et al., 2023). Under optimized parameters (impedance $Z=1$–$2\,\mathrm{k}\Omega$, $E_J\approx E_L$, $f_{01}=1\,\mathrm{GHz}$), single-qubit average gate fidelities $F>99.99\%$ are predicted even with current material quality (Duda et al., 28 Apr 2025).

6. Design Optimization and Scalability

Design optimization leverages the parameter space of $(E_J, E_L, E_C)$. At $\Phi = \Phi_0/2$, $E_L = E_J$ cancels the quadratic term, leaving a pure quartic (Duffing) confining potential and maximizing anharmonicity (Duda et al., 28 Apr 2025). The choice of impedance $Z=\sqrt{L/C}$ influences both relative flux curvature and sensitivity to dielectric loss: $Z=1$–$2\,\mathrm{k}\Omega$ balances low dephasing with fast gates.

IST devices employ compact thin-film kinetic inductors (Al, NbTiN), fully compatible with conventional Josephson junction processes and enabling tileable, high-density layouts. Weak dependence on flux and charge allows for frequency crowding mitigation and straightforward multi-qubit lattices (Fasciati et al., 2024, Kalacheva et al., 2023). Experimental designs demonstrate that the added inductance does not degrade coherence and in some cases can increase anharmonicity by factors of $1.5$–$2$ compared to bare transmons (Kalacheva et al., 2023).

A comparative summary of key modalities (values are typical; see references for full device parameter sets):

Qubit Type	$f_{01}$ (GHz)	$\alpha/2\pi$ (MHz)	$T_1$ ($\mu$s)	Flux Dispersion (MHz)
Transmon	$4.5$	$-250$	$10^2$–$10^3$	$>100$
IST (unimon, kinemon)	$1$–$7$	$-50$ to $+300$	$9$–$20$	$<5$
Fluxonium	$0.6$	$\gg10^4$	$>10^3$	$\sim1$

7. Advanced Architectures and Applications

IST circuits enable new physical regimes and protocols:

• Protected Qubit Encodings: The coexistence of protected fluxon levels (degenerate double-well minima) with robust plasmon transitions enables schemes that leverage long-lived memory and fast gates in the same hardware (Hassani et al., 2022).
• QND Readout: Tunable longitudinal coupling ($\hat{\sigma}_z \hat{a}^\dagger\hat{a}$) allows fast, dispersive, strictly QND measurement, and is foundational for scalable error-corrected architectures (Richer et al., 2017, Richer, 2018).
• Suppression of Parasitic Interactions: Dynamic tuning of anharmonicities allows for suppression of unwanted $ZZ$ cross-talk and the realization of strictly local gates.
• Strong Parametric Drives: ISTs allow multi-wave mixing, parametric amplification, and Hamiltonian exploration inaccessible to bare transmons due to their instability (Verney et al., 2018).
• Bosonic Codes and Reservoir Engineering: The high nonlinearities, low leakage, and robust phase confinement make the IST well suited for cat codes, bosonic encodings, and engineered dissipation protocols.

As a generalized platform, the inductively shunted transmon unifies the beneficial aspects of transmon, fluxonium, and flux qubits, presenting a versatile and robust qubit modality for next-generation cQED, quantum information processing, and quantum simulation (Hassani et al., 2022, Duda et al., 28 Apr 2025, Kalacheva et al., 2023, Dumur et al., 2015).