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arXiv:2603.06482 · Physical review. B./Physical review. B

Parity readout in Majorana box qubits from the dispersive to the resonant regime

Sara M. Benjadi, Reinhold Egger

Published Jun 12, 2026• 0 citations
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Abstract

We study theoretical models for charge reflectometry and capacitive readout of the Majorana parity degree of freedom in Majorana box qubits, considering decoherence channels within the framework of the Lindblad master equation. Noting that a parity-dependent dynamical susceptibility χ z ( ω ) governs both readout schemes, we provide a general expression for χ z ( ω ) which covers the full crossover from the resonant regime to the off-resonant dispersive regime. In addition, we reexamine previous results which were obtained under a semiclassical factorization assumption. Using three different error measures, we show that this approximation is quantitatively justified in the dispersive regime. In the resonant regime, however, we find deviations from exact reference data, obtained by numerical solution for the steady state of the full Lindblad equation. These deviations are typically of the order of a few percent in the considered error measures.

DOI
10.1103/j2x7-1h9c

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We study theoretical models for charge reflectometry and capacitive readout of the Majorana parity degree of freedom in Majorana box qubits, considering decoherence channels within the framework of the Lindblad master equation. Noting that a parity-dependent dynamical susceptibility χ z ( ω ) governs both readout schemes, we provide a general expression for χ z ( ω ) which covers the full crossover from the resonant regime to the off-resonant dispersive regime. In addition, we reexamine previous results which were obtained under a semiclassical factorization assumption. Using three different error measures, we show that this approximation is quantitatively justified in the dispersive regime. In the resonant regime, however, we find deviations from exact reference data, obtained by numerical solution for the steady state of the full Lindblad equation. These deviations are typically of the order of a few percent in the considered error measures.
We analyze a readout scheme for Majorana qubits based on dispersive coupling to a resonator.We consider two variants of Majorana qubits: the Majorana transmon and the Majorana box qubit.In both cases, the qubit-resonator interaction can produce sizeable dispersive shifts in the megahertz range for reasonable system parameters, allowing for submicrosecond readout with high fidelity.For Majorana transmons, the light-matter interaction used for readout manifestly conserves Majorana parity, which leads to a notion of quantum nondemolition (QND) readout that is stronger than for conventional charge qubits.In contrast, Majorana box qubits only recover an approximately QND readout mechanism in the dispersive limit where the resonator detuning is large.We also compare dispersive readout to longitudinal readout for the Majorana box qubit.We show that the latter gives faster and higher fidelity readout for reasonable parameters, while having the additional advantage of being manifestly QND, and so may prove to be a better readout mechanism for these systems.
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We present a theoretical analysis of a Majorana-based qubit consisting of two topological superconducting islands connected via a Josephson junction. The qubit is operated by electrostatic gates which control the coupling of two of the four Majorana zero modes. At the end of the operation, readout is performed in the charge basis. Even though the operations are not topologically protected, the proposed experiment can potentially shed light on the coherence of the parity degree of freedom in Majorana devices and serve as a first step towards topological Majorana qubits. We discuss in detail the charge-stability diagram and its use for characterizing the parameters of the devices, including the overlap of the Majorana edge states. We describe the multilevel spectral properties of the system and present a detailed study of its controlled coherent oscillations, as well as decoherence resulting from coupling to a non-Markovian environment. In particular, we study a gate-controlled protocol where conversion between Coulomb-blockade and transmon regimes generates coherent oscillations of the qubit state due to the overlap of Majorana modes. We show that, in addition to fluctuations of the Majorana coupling, considerable measurement errors may be accumulated during the conversion intervals when electrostatic fluctuations in the superconducting islands are present. These results are also relevant for several proposed implementations of topological qubits which rely on readout based on charge detection.
Solid-state qubits incorporating quantum dots can be read out dispersively. Here, we theoretically describe physical mechanisms that render such reflectometry-based readout schemes imperfect. We discuss charge qubits, singlet-triplet spin qubits, and Majorana qubits. In our model, we account for readout errors due to slow charge noise, and due to overdriving, when the large amplitude of the probe field generates errors. A key result is that for charge and spin qubits, the readout fidelity saturates at large probe strengths, whereas for Majorana qubits, there is an optimal probe strength which provides a maximized readout fidelity. We also point out the existence of severe readout errors appearing in a resonancelike fashion as the pulse strength is increased, and show that these errors are related to probe-induced multiphoton transitions. Besides providing practical guidelines toward optimized readout, our study might also inspire ways to use gate reflectometry for device characterization.
In this work we revisit the problem of a Majorana box qubit (MBQ) readout by a point-contact (PC) detector. The logic states of the MBQ are associated with the combined fermion parities of the MBQ and its tunnel-coupled quantum dot, which is measured by a PC detector. Beyond the existing studies on limiting bias voltage regimes, we analyze the steady-state current and the current power spectrum across all bias voltages. Our results indicate that the MBQ readout via the parity-dependent detector current is effective only at low bias voltage regime and requires the dot energy level to be off-resonance with the Majorana qubit. In contrast, the current power spectrum allows MBQ readout through the parity-dependent Rabi oscillation peak signals for arbitrary bias voltages, without restrictions on the dot energy level. Particularly, with focus on the MBQ measurement visibility, we analyze the peak-to-pedestal ratio for each characteristic peak (associated with each logic state of the qubit) and the signal-to-noise ratio of the two peaks. By examining these two metrics, we identify the optimal bias voltage window for the PC detector at low temperature limit.