Scaling and full characterization of the fully frustrated Josephson-junction ladder superinductor

Bell, Sadovskyy, Ioffe, Kitaev, and Gershenson (Phys. Rev. Lett. 109, 137003 (2012)) demonstrated a superinductor — a dissipationless circuit element whose microwave impedance exceeds the resistance quantum — built as a ladder of nanoscale Josephson junctions frustrated by half a flux quantum, achieving inductances up to 3 µH and strong, flux-tunable quartic nonlinearity with phase slips exponentially suppressed. The demonstration left open the maturation of this element: parasitic-capacitance reduction toward ~1 fF and impedances of tens of kΩ at GHz frequencies, direct measurement of the internal-mode spectrum at full frustration, isolation of the intrinsic decoherence rate from readout coupling, quantitative quantum modeling of ladders too long for exact diagonalization, and integration into protected-qubit circuits and adiabatic switches. None of these ladder-specific questions has been closed in the literature to date: later high-impedance elements (geometric, granular-aluminum, nanowire, and stacked-junction superinductors) and protected-qubit experiments (rhombus chains, bifluxon, protected fluxonium) use other circuit implementations, while the ladder's scaling, internal-mode characterization, decoherence budget, large-N modeling, and architectural use remain unrealized.

Background

A superinductor is a superconducting circuit element whose characteristic impedance $Z=\sqrt{L/C}$, with $L$ its inductance and $C$ its capacitance to ground, greatly exceeds the resistance quantum $R_{Q}=h/(2e)^{2}\approx 6.45$ kΩ (the original paper writes $h/4e^{2}$) while dissipating essentially no power. Impedances well above $R_{Q}$ suppress the zero-point charge fluctuations of a shunted island, which is why such elements are wanted for superconducting qubits protected from charge noise, for fault-tolerant qubit proposals, and for isolating small Josephson junctions so that Bloch oscillations — the dual of Josephson oscillations, in which a voltage bias produces a periodic current — could define a quantum current standard. Ordinary geometrical inductance cannot reach this regime because the magnetic energy of a wire loop is unavoidably accompanied by a parasitic capacitance that bounds $Z$ by roughly the fine-structure-constant fraction $\alpha R_{Q}$.

The alternatives are kinetic inductance, associated with the inertia of the Cooper-pair condensate, of either disordered thin films or Josephson-junction arrays; the fluxonium qubit (Manucharyan, Koch, Glazman, and Devoret, Science 326, 113 (2009)) already used a linear array of Josephson junctions as a ~nH-scale superinductor. A Josephson junction is a tunnel contact between superconductors, characterized by its Josephson energy $E_{J}$ (the coupling of the superconducting phases across the barrier) and its charging energy $E_{C}=e^{2}/2C$ (the energy cost of moving one electron onto the junction capacitance $C$). Large inductance requires small $E_{J}$, but then quantum phase slips — tunneling events of the superconducting phase between neighboring wells of the Josephson potential, whose rate grows as $\exp[-(8E_{J}/E_{C})^{1/2}]$ — cause dissipation and, together with fluctuating offset charges via the Aharonov–Casher interference of flux tunneling paths, decohere the circuit. Linear chains of junctions with large $E_{J}/E_{C}$ avoid phase slips but are essentially linear elements whose impedance is limited by their own parasitic capacitance, and frustrated chains of dc SQUIDs (superconducting loops interrupted by two junctions, threaded by magnetic flux) suffer strongly enhanced phase-slip rates.

Bell, Sadovskyy, Ioffe, Kitaev, and Gershenson (Phys. Rev. Lett. 109, 137003 (2012)) proposed and demonstrated a way around this trade-off: a ladder of unit cells, each an asymmetric dc-SQUID with one small junction (Josephson energy $E_{JS}$) and three large junctions (energy $E_{JL}$), all cells threaded by the same magnetic flux $\Phi$. At full frustration $\Phi=\Phi_{0}/2$ ($\Phi_{0}=h/2e$ is the flux quantum) and at the optimal junction-energy ratio $r=E_{JL}/E_{JS}=r_{0}$ ($r_{0}\approx 4.1$–$4.5$ for ladders of $N=6$ and 24 cells, renormalized downward by quantum fluctuations by $\delta r\sim 0.5$–$0.7$), the curvature of the Josephson energy near zero phase vanishes: the effective kinetic inductance diverges, the element becomes strongly nonlinear with a tunable quartic anharmonicity, and — because the energy is an even function of the flux — the device is first-order insensitive to flux noise, all while phase slips stay exponentially suppressed in the large junctions with $E_{JL}/E_{CL}\approx 100$ ($E_{CL}$ the charging energy of the large junctions). For a 24-cell device the measured inductance at full frustration reached 3 µH (impedance an order of magnitude above $R_{Q}$), with Rabi decay times of a superinductor-based qubit exceeding 1 µs, limited by intentional readout coupling.

The authors projected that meandering the ladder and shrinking the 3 × 5 µm² unit cell would reduce the parasitic capacitance from ~3 fF to ~1 fF, giving an impedance approaching 50 kΩ at 3 GHz, and envisioned applications to qubits protected from both flux and charge noise, to Kitaev's current-mirror protected qubit (arXiv:cond-mat/0609441) and related protected-qubit architectures reviewed by Douçot and Ioffe (Rep. Prog. Phys. 75, 072001 (2012)), and to an adiabatic switch — the superinductor combined with a small junction — whose impedance changes exponentially with magnetic flux, which protected-gate schemes require.

The 2012 experiment verified the lowest ('global') mode of a 6-cell ladder against exact diagonalization of the circuit Hamiltonian, but for the 24-cell ladder only a quasiclassical approximation (treating the junction phases as classical variables, corrected for quantum renormalization of $r_{0}$) was available, the internal excitation frequencies $\omega_{\text{int}}$ were estimated (above ~10 GHz at full frustration) but not measured, and the intrinsic decoherence rate was only argued to be small. These gaps define the problem; no work is known to date that has closed them for the frustrated ladder, although alternative superinductor implementations — geometric superinductors (Peruzzo et al., Phys. Rev. Applied 14, 044055 (2020)), stacked-junction hyperinductances (arXiv:2505.02764), SQUID-free Fraunhofer-tuned arrays (arXiv:2210.12119), granular-aluminum and nanowire kinetic inductors, and a high-temperature-superconductor superinductor (Nat. Mater. 2025) — have advanced the shared goal of high impedance with low parasitic capacitance by other mechanisms.

Problem Statement

Can the fully frustrated Josephson-junction ladder superinductor of Bell et al. (2012) be established as a scalable, quantitatively understood, low-decoherence circuit element and used as such in protected superconducting-qubit circuits? Concretely, the following interlinked questions remain open for this element. (1) Scaling and parasitics: fabricate and verify long ladders with the parasitic capacitance reduced to the ~1 fF level, so that the microwave impedance reaches the tens-of-kΩ range at GHz frequencies, and establish the quantitative design rules (junction-energy ratio $r$ relative to the optimum $r_{0}$, cell count $N$, junction sizes) governing this scaling. (2) Internal modes: directly measure the ladder's internal excitation spectrum at full frustration — the source only estimates $\omega_{\text{int}}/2\pi>10$ GHz for the 24-cell device — and determine how these internal modes couple to and limit the 'global' mode that applications require to be well separated from them. (3) Intrinsic decoherence: measure the intrinsic energy-relaxation and dephasing rates of the ladder decoupled from intentional readout coupling, with phase slips exponentially suppressed, and establish how these rates scale with $N$, $r-r_{0}$, and the junction parameters $E_{J}/E_{C}$. (4) Quantum modeling: develop and validate a quantitative quantum treatment of long ladders ($N\gtrsim 24$) for which exact diagonalization is infeasible, testing the quasiclassical approximation used in the source. (5) Applications: demonstrate, in circuits actually built from this ladder element, the proposed uses — qubits protected against both flux and charge noise, flux-controlled controllable coupling between qubits, and the adiabatic switch whose impedance changes exponentially with magnetic flux as required for protected-gate and current-mirror-qubit architectures (Kitaev, arXiv:cond-mat/0609441; Douçot and Ioffe, Rep. Prog. Phys. 75, 072001 (2012)) — including the gate operations and error suppression these architectures promise. These five are complementary components of a single maturation objective, not alternative routes to it: items (1)–(4) mature the element and item (5) demonstrates it in a protected circuit.

References

1. M. T. Bell, I. A. Sadovskyy, L. B. Ioffe, A. Yu. Kitaev, and M. E. Gershenson, 'Quantum Superinductor with Tunable Nonlinearity,' Phys. Rev. Lett. 109, 137003 (2012). DOI: 10.1103/PhysRevLett.109.137003; arXiv:1206.0307, https://arxiv.org/abs/1206.0307
2. A. Yu. Kitaev, 'Protected qubit based on a superconducting current mirror,' arXiv:cond-mat/0609441 (2006), https://arxiv.org/abs/cond-mat/0609441
3. B. Douçot and L. B. Ioffe, 'Physical implementation of protected qubits,' Rep. Prog. Phys. 75, 072001 (2012). DOI: 10.1088/0034-4885/75/7/072001, https://doi.org/10.1088/0034-4885/75/7/072001
4. V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, 'Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets,' Science 326, 113 (2009). DOI: 10.1126/science.1175552, https://doi.org/10.1126/science.1175552
5. M. T. Bell, J. Paramanandam, L. B. Ioffe, and M. E. Gershenson, 'Protected Josephson Rhombus Chains,' Phys. Rev. Lett. 112, 167001 (2014). DOI: 10.1103/PhysRevLett.112.167001; arXiv:1311.6521, https://arxiv.org/abs/1311.6521
6. K. Kalashnikov, W. Hsieh, W. Zhang, W. Lu, P. Kamenov, A. Di Paolo, A. Blais, M. Gershenson, and M. Bell, 'Bifluxon: Fluxon-Parity-Protected Superconducting Qubit,' PRX Quantum 1, 010307 (2020). DOI: 10.1103/PRXQuantum.1.010307; arXiv:1910.03769, https://arxiv.org/abs/1910.03769
7. D. K. Weiss, A. C. Y. Li, D. G. Ferguson, and J. Koch, 'Spectrum and coherence properties of the current-mirror qubit,' Phys. Rev. B 100, 224507 (2019). DOI: 10.1103/PhysRevB.100.224507, https://doi.org/10.1103/PhysRevB.100.224507
8. M. Peruzzo, A. Trioni, F. Hassani, M. Žemlička, and J. M. Fink, 'Surpassing the Resistance Quantum with a Geometric Superinductor,' Phys. Rev. Applied 14, 044055 (2020). DOI: 10.1103/PhysRevApplied.14.044055; arXiv:2007.01644, https://arxiv.org/abs/2007.01644
9. P. Manset, J. Palomo, A. Schmitt, K. Gerashchenko, R. Rousseau, H. Patange, P. Abgrall, E. Flurin, S. Deléglise, T. Jacqmin, and L. Balembois, 'Hyperinductance based on stacked Josephson junctions,' arXiv:2505.02764 (2025), https://arxiv.org/abs/2505.02764
10. R. Kuzmin, N. Mehta, N. Grabon, and V. E. Manucharyan, 'Tuning the inductance of Josephson junction arrays without SQUIDs,' arXiv:2210.12119 (2022), https://arxiv.org/abs/2210.12119
11. N. Crescini, S. Cailleaux, W. Guichard, C. Naud, O. Buisson, K. Murch, and N. Roch, 'Evidence of dual Shapiro steps in a Josephson junction array,' Nat. Phys. (2023). DOI: 10.1038/s41567-023-01961-4; arXiv:2207.09381, https://arxiv.org/abs/2207.09381
12. A. Gyenis, A. Di Paolo, J. Koch, A. Blais, A. Houck, and D. I. Schuster, 'Moving beyond the Transmon: Noise-Protected Superconducting Quantum Circuits,' PRX Quantum 2, 030101 (2021). DOI: 10.1103/PRXQuantum.2.030101; arXiv:2106.10296, https://arxiv.org/abs/2106.10296
13. A. J. Kerman, 'Efficient numerical simulation of complex Josephson quantum circuits,' arXiv:2010.14929 (2020), https://arxiv.org/abs/2010.14929
14. W. Ardati, S. Léger, S. Kumar, V. Suresh, D. Nicolas, C. Mori, F. D'Esposito, T. Vakhtel, O. Buisson, Q. Ficheux, and N. Roch, 'Using Bifluxon Tunneling to Protect the Fluxonium Qubit,' Phys. Rev. X 14, 041014 (2024). DOI: 10.1103/PhysRevX.14.041014, https://doi.org/10.1103/PhysRevX.14.041014
15. P. Aramburu Sanchez, T. F. Q. Larson, A. P. McFadden, C. Schrade, J. Combes, and A. Gyenis, 'Revisiting the multi-mode rhombus circuit as a biased-noise qubit,' arXiv:2605.06430 (2026), https://arxiv.org/abs/2605.06430
16. Y. K. Srivastava, T. C. I. Pang, M. Gupta, M. Manjappa, P. Agarwal, J. Lesueur, and R. Singh, 'YBa2Cu3O7 as a high-temperature superinductor,' Nat. Mater. (2025). DOI: 10.1038/s41563-024-02107-4, https://doi.org/10.1038/s41563-024-02107-4