Can an Inductive Shunt Stabilize Measurement-Induced State Transitions in Transmons?

Offset charge makes transmon MIST locations drift

Faster dispersive readout needs more photons in the readout resonator, and more photons create unwanted resonances in the joint qubit-resonator system. Those resonances drive measurement-induced state transitions (MIST), which can push the qubit into a high energy state that is hard to revert and damaging for quantum error correction.

The specific problem this paper targets is not only that MIST exists, but that in transmon qubits the locations in readout parameter space where MIST occurs depend strongly on offset charge. Because offset charge fluctuates in time, those MIST locations drift, which degrades error correction performance. Prior transmon strategies avoided the drift by using very large qubit-resonator detunings to remove MIST entirely, or by stabilizing the offset charge. This work takes a different route: adding an inductive shunt to the transmon, which provides a dc short across the superconducting islands and removes the dependence on static charge offsets.

The inductive shunt removes dc offset-charge dependence

The inductively-shunted transmon (IST) sits between the transmon and the fluxonium. Like the transmon, it has a large capacitance that eases integration into large qubit arrays and a single-well potential that permits analytic models. Like the fluxonium, the inductive shunt removes dc offset-charge dependence, and the anharmonicity tunes over a wide range. The paper explores the regime where the charging energy is much smaller than the inductive and Josephson energies, which are comparable.

This distinction matters for how MIST must be modeled. In a transmon, charge matrix elements shrink rapidly for states above the edge of the cosine potential, so processes where the qubit absorbs many resonator photons are suppressed, and treating the resonator as a classical drive works. In the IST, the unbounded potential energy keeps charge matrix elements large for all levels, so the resonator cannot be approximated as a drive, and MIST cannot be described as an ionization process.

Two IST designs cover both sides of the resonator

Because MIST depends on whether the qubit frequency is above or below the readout resonator frequency, the device was designed with two groups of six flux-tunable qubits. The first group was designed so readout works well when the qubit frequency exceeds the resonator frequency; the second group was designed for the opposite case. Both groups chose a qubit-resonator coupling giving a dispersive shift comparable to the resonator linewidth at the intended flux-insensitive point, and each group is read out through a shared Purcell filter and readout line.

The qubits and resonators are fabricated on separate chips that are bump bonded, with capacitive coupling through the gap between chips. The shunt inductance for each qubit is a linear array of 9 Josephson junctions, each roughly 9 times larger in area than the small junction, all fabricated in a single lithography step at a critical current density of 22.5 A cm^-2, corresponding to a junction plasma frequency of 18.6 GHz. The authors estimate phase-slip rates for a single array junction of 0.2 Hz in the first group and 0.5 mHz in the second.

An effective Kerr model predicts the measured spectra

The full-circuit model treats the junction array as a linear inductance and includes the qubit's capacitive, Josephson, and inductive energies, the readout resonator, and a capacitive qubit-resonator coupling. Because that complete model is computationally expensive, the authors derive a simplified effective model in which the IST behaves as a weakly anharmonic oscillator, with a dressed frequency and anharmonicity linked to circuit parameters through an effective IST-mode impedance.

They use this effective model to predict transition frequencies at different flux biases during calibration and to seed a full numerical fit of spectroscopy data. To measure the device parameters, they calibrate a readout that discriminates states |0>, |1>, and |2>, then perform spectroscopy: dynamically bias the qubit to a spectroscopy bias, wait 2 microseconds for the frequency to settle, apply a 300 ns variable-frequency tone, return to the idle bias, and record the frequency giving the highest |1> population as omega_10. They constrain the model further by measuring omega_21 with an additional |0>-to-|1> pulse before the spectroscopy pulse, and estimate the resonator frequency versus qubit bias from the maximum phase gradient of the readout tone across the resonator. The measured spectra were well predicted by the approximate Kerr Hamiltonian.

Renormalization is essential for IST MIST predictions

The paper compares three descriptions of the qubit-resonator spectrum: a full quantum model, a renormalized semiclassical model, and a simple semiclassical model. For an IST qubit, the renormalized semiclassical approximation recreates the quantum crossing at about 15 photons, while the simple semiclassical model fails and predicts a crossing at about 5 photons. For a transmon, both the renormalized and simple models predict crossings much closer to the quantum value of about 15 photons, though the renormalized model still agrees better.

The authors conclude that renormalized semiclassical approximations must be used for ISTs and other qubits with similar energy structure, such as fluxonium, otherwise MIST predictions will be inaccurate. This is a modeling requirement, not a device result: it follows from the unbounded IST potential keeping charge matrix elements large, which breaks the driven-resonator approximation that works for transmons.

IST MIST features are qualitatively more stable than transmon

The authors characterized MIST in ISTs over a broad flux range and found agreement with full quantum simulations and with the renormalized semiclassical model. They also observed qualitatively more stable MIST features in the IST compared with a similar transmon in a Willow processor, consistent with the insensitivity to dc offset charge. The main features of MIST were captured by the renormalized semiclassical model without needing to include effects of resonances in the junction chain.

The paper reports T1 times up to 100 microseconds in these qubits, with the highest values near 100 microseconds. However, each qubit has a large number of strongly coupled resonant defects, which makes running the experiments difficult. In the excited-state MIST experiment, the data are hard to interpret in general because resonant defects and Purcell loss both induce qubit decay: if T1 during readout is on the order of the readout time, the qubit may decay to |0> and then MIST might occur. Some data-model discrepancies are expected from this effect. For example, the missing feature near 3 GHz in the semiclassical model is due to T1 decay from the Purcell effect not being included in that model; the full simulation catches this feature, but does not predict its onset photon number accurately because it lacks an accurate description of the Purcell decay frequency dependence. Significant |2> leakage appeared irrespective of readout power at some flux biases, which the authors attribute to stray drives on the qubit from electronics and to the resonant defects.

Stable MIST landscapes simplify readout optimization

The paper's practical claim is that ISTs are promising candidates for quantum error correction because they combine large anharmonicity at low frequency with no offset charge noise, large capacitance for coupling to neighboring qubits, and a stable MIST landscape that simplifies readout optimization. The stability is the key difference from the transmon: instead of avoiding MIST by using very large qubit-resonator detunings or by actively stabilizing offset charge, the inductive shunt removes the offset-charge dependence that makes transmon MIST locations drift.

The authors expect these modeling techniques to allow design of highly performant readout circuits in these qubits that exceed the stability of transmon readout, and they state that the results extend to other inductively-shunted qubits. The most decision-relevant comparison for a reader is therefore between two ways of handling MIST: transmon approaches that remove MIST with very large detunings or stabilize offset charge, versus an IST whose MIST landscape is stable in time but whose modeling requires the renormalized semiclassical treatment rather than the simple driven-resonator approximation.

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