Abstract
Protecting qubits from noise is essential for building reliable quantum computers. Topological qubits offer a route to this goal by encoding quantum information non-locally, using pairs of Majorana zero modes. These modes form a shared fermionic state whose occupation—either even or odd—defines the fermionic parity that encodes the qubit1. Notably, this parity can only be accessed by a measurement that couples two Majoranas to each other. A promising platform for realizing such qubits is the Kitaev chain1, implemented in quantum dots coupled using superconductors2. Even the minimal two-site chain hosts a pair of Majorana modes, often called ‘poor man’s Majoranas’, which are spatially separated but offer limited protection compared with longer chains3,4,5. Here we introduce a measurement technique that reads out their parity through quantum capacitance. Our method couples two Majoranas and resolves their parity in real time, visible as random telegraph switching with lifetimes exceeding a millisecond. Simultaneous charge sensing confirms that the two parity states are charge neutral and remain indistinguishable to a probe that does not couple the modes. These results establish the essential readout step for time-domain control of Majorana qubits, resolving a long-standing experimental challenge.
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Data availability
The raw data measured on the device presented in this work are available at https://doi.org/10.4121/227fd419-fded-4a96-ab62-421a0cd57fa5 (ref. 51).
Code availability
The data processing and plotting code and the code used for the theory calculations are available at https://doi.org/10.4121/227fd419-fded-4a96-ab62-421a0cd57fa5 (ref. 51).
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Acknowledgements
This work has been supported by the Dutch Organization for Scientific Research (NWO), Microsoft Corporation Station Q, the Spanish Ministry of Science (grant no. PID2021-125343NB-I00) and the Horizon Europe framework programme of the European Commission through the European Innovation Council Pathfinder grant no. 101115315 (QuKiT). We thank O. W. B. Benningshof and J. D. Mensingh for technical assistance with the cryogenic electronics, F. K. Malinowski for contributing to the design of the experiment, C. X. Liu and S. L. D. ten Haaf for their input on the manuscript, T. Dvir, A. Lombardi, V. P. M. Sietses, F. J. Bennebroek Evertsz, M. Wimmer, J. D. Torres Luna, S. Miles, S. Goswami, S. Roelofs, D. Joshi and A. R. Akhmerov for discussions, J. M. Hornibrook and D. J. Reilly for providing the frequency multiplexing chips and S. Gazibegovic for contribution to nanowire growth.
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Contributions
The sample was fabricated by N.v.L. and B.R. Measurements were performed by N.v.L., F.Z. and T.V.C. The experiment was designed by N.v.L., F.Z. and G.W. The data were analysed by N.v.L. and F.Z. The experimental set-up was designed and implemented by N.v.L., F.Z., B.R., G.W. and D.v.D. The manuscript was prepared by N.v.L. and F.Z., with input from all authors. G.W., A.B., D.v.D., Y.Z., W.D.H. and G.P.M. contributed to understanding and interpretation of the data through regular discussions. The project was supervised by L.P.K. Modelling and simulations of the system were carried out by G.O.S. and R.A. The InSb nanowire growth was performed by G.B. and E.P.A.M.B.
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Extended data figures and tables
Extended Data Fig. 1 Measurement set-up.
Detailed wiring diagram for the DC and RF lines used to measure the device. To reduce the generation of non-equilibrium quasiparticles in the superconducting film, we use infrared filters for the RF lines. Furthermore, light-tight packaging encloses the device. Finally, we note that an extra resonator was connected to the left normal lead but it was not used.
Extended Data Fig. 2 Shadow lithography with masks grown on-chip.
a, False-coloured tilted scanning electron micrograph of a nanowire (InSb, shown in green) on top of the gate electrode array (Pd, shown in red) after deposition of the superconductor (Al, shown in blue). The shadow mask (HSQ, shown in purple) has a narrow slit in the centre, through which the Al strip that contacts the nanowire is evaporated, at an angle of 30° with respect to the substrate. A star is placed at approximately the same location in this and the other panels to help identify the orientation of the substrate. b, False-coloured top-view scanning electron micrograph of the same nanowire as in panel a, after the shadow mask has been mechanically removed using a nanomanipulator. c, Optical image of the PMMA mask that covers the nanowire and three leads towards the device. Image taken after patterning and development of the mask but before etching of the Al film using Transene type D. d, Optical image of the device after etching of the Al film and stripping of the protective mask. The central Al lead connects the nanowire to a W pad, which facilitates wire bonding (not shown here).
Extended Data Fig. 3 Evolution of quantum capacitance with hybrid gate voltage VH.
a–c, Quantum capacitance charge-stability diagrams measured using the middle resonator while the device is isolated from the normal leads. Each panel corresponds to a different setting of the hybrid gate voltage VH, which tunes the energy of the ABS in the superconducting segment. As VH increases, we observe a progressive weakening of the signal along the interdot detuning axis. This behaviour is attributed to an increase in the strength of CAR between the QDs, mediated by the lowering of the ABS energy. Because quantum capacitance is inversely proportional to CAR strength, a stronger CAR leads to reduced signal contrast. This trend provides a qualitative probe of the CAR amplitude and helps identify regions in which t ≈ Δ, as required for the PMM sweet spot.
Extended Data Fig. 4 Detailed analysis of time-trace performance at the PMM sweet spot.
a,b, Signal-to-noise ratio (SNRM, panel a) and parity lifetimes (panel b) as a function of readout amplitude. All data were taken with the same gate settings and 150-μs integration time as in Fig. 3d. The dotted grey line indicates the amplitude used for Figs. 3 and 4. Beyond a certain readout amplitude, the SNR decreases and the lifetimes become parity-dependent, consistent with drive-induced modulation along the common-mode axis δ that preferentially stabilizes the even ground state. This observation suggests that the system can be initialized in the even state by driving the middle resonator with a high amplitude. The time traces in panels a and b were averaged in time bins of 160 μs (instead of 150 μs) to ensure that the averaging time is an exact multiple of the sampling rate (50 kHz). c,d, SNRM (c) and readout error (d) of the time trace presented in Fig. 3 versus integration time. The readout error was estimated as \([1-{{\rm{e}}}^{-{\tau }_{{\rm{bin}}}/{\tau }_{{\rm{avg}}}}{\rm{erf}}({{\rm{SNR}}}_{{\rm{M}}}({\tau }_{{\rm{bin}}})/\sqrt{2})]/2\) (ref. 23), in which τbin is the integration time and τavg = 1.85 ± 0.03 ms. The dotted grey line indicates the integration time τbin = 150 μs used for all of the reported measurements unless otherwise specified. e, Histograms of dwell times in the even and odd parity states, normalized by total counts. The dotted lines show fits with the corresponding Poissonian probability densities \({{\rm{e}}}^{-t/{\tau }_{{\rm{o,e}}}}/{\tau }_{{\rm{o,e}}}\), confirming that the dwell times follow an exponential distribution. τo,e have been estimated as the mean of the dwell times in the odd and even parity states. Their uncertainty has been estimated as the standard deviation of the mean. f, Average switching time, τavg, estimated for varying integration time. As the integration time increases, the estimation of τavg is systematically biased towards higher values. See Methods section ‘Time traces measurement and analysis’ for a more detailed discussion. g, Power spectral density (PSD) of the time trace using an integration time of 10 μs. By fitting it with a Lorentzian model, we extract an average switching time of 1.51 ± 0.07 ms. The discrepancy between this estimate and those obtained with a hidden Markov model or exponential fitting further suggests that the long integration time needed to obtain high SNR could systematically bias the estimation of τavg. See Methods section ‘Time traces measurement and analysis’ for a more detailed discussion. h,i, Analysis of the time trace of Fig. 3c using the analysis methods and code adapted from ref. 23. In panel h, the histogram is fit to a weighted sum of Gaussian distributions. The extracted SNR = d/(σe + σo) = 1.93 is comparable with the hidden Markov model analysis. In panel i, the dwell time distributions are extracted using a Gaussian mixture model and a thresholding algorithm. The parity lifetimes are then estimated by fitting the distributions with an exponential model. The estimates are comparable with those obtained with the hidden Markov model analysis.
Extended Data Fig. 5 Evolution of charge-stability diagrams and switching behaviour across the ECT-to-CAR transition.
a–d, Measurements at VH = 1.685 V, in which t > Δ. The charge-stability diagram measured by quantum capacitance (a) shows strong signal along the detuning axis. The dotted grey dotted box marks the region in which 10-s time traces were measured to estimate (Methods section ‘Time traces measurement and analysis’) the signal-to-noise ratio (SNRM, panel b), parity polarization (PM, panel c) and average switching time (τavg, panel d). In these panels, the grey contours indicate where SNRM = 0.5. In panel c, a dashed black line highlights the contour at which PM = 0, calculated after applying a Gaussian filter with standard deviation of 1 pixel to reduce noise. This contour resembles an avoided crossing along the detuning axis, characteristic of the condition t > Δ. e–h, Same measurements at VH = 1.665 V, in which t ≈ Δ near the PMM sweet spot. In this case, the parity polarization map (panel g) shows no avoided crossing. Instead, regions of positive and negative parity polarization alternate around the centre of the charge-stability diagram, indicating that CAR and ECT are balanced. This is the dataset used in the main text. i–l, Same at VH = 1.645 V, at which CAR dominates (t < Δ). Here the polarization map (panel k) shows that the avoided crossing reappears but now along the common-mode axis, reversing the pattern seen in c. This indicates that the system transitioned to a regime in which t < Δ. m–p, Corresponding data from panels i–l using the CS, recorded by the right resonator. The SNRR vanishes at the centre of the charge-stability diagram, when the QDL is on resonance (VLD ≈ 264 mV). This indicates that a charge measurement of QDR is not sufficient to determine the parity of the system. q, Map of the Pearson correlation coefficient ρMR between quantum capacitance and CS signals in the t < Δ regime. The sign change of ρMR (black arrows) no longer follows a horizontal line. This is consistent with CAR dominating over ECT, as explained in more detail in Methods section ‘Model’.
Extended Data Fig. 6 Transport measurement of the PMM sweet spot using RF reflectometry.
a–c, Zero-bias charge-stability diagrams for different values of VH, used to tune the relative strength of ECT and CAR. In all panels, the left lead is operated as a tunnel probe, whereas the right side is blocked by the CS. As a result, transport occurs predominantly through QDL and visibility of the QDR resonance is suppressed. The orientation of the avoided crossing in panel a indicates that t < Δ. As VH increases, the avoided crossing first disappears when the sweet spot t ≈ Δ (b) is approached. Further increasing VH opens the avoided crossing in the opposite direction, indicating that t > Δ (c). The dotted lines in panel b indicate the axes along which the spectra of panels e–h were measured. d, Tunnel spectroscopy varying \({V}_{{\rm{L}}}^{{\rm{DC}}}\) at the centre of the charge-stability diagram of panel b. The spectrum exhibits a zero-bias peak separated from two excited states by a gap of about 30 μeV, consistent with PMMs in a minimal Kitaev chain. This linecut was extracted along the dotted grey dotted line of panel e. e,f, Spectra measured while varying \({V}_{{\rm{L}}}^{{\rm{DC}}}\) and the electrochemical potential of the left (e) or right (f) QD. The stable zero-bias peak indicates robustness against local perturbations. g,h, Spectra measured while varying \({V}_{{\rm{L}}}^{{\rm{DC}}}\) and the electrochemical potential of both QDs along the common-mode (g) and detuning (h) axis. The zero-bias peak splits quadratically, illustrating limited protection against global perturbations.
Extended Data Fig. 7 Parity readout using a nearby CS.
a, Illustration showing that a probe that does not couple the two Majoranas to each other cannot access the parity iγ1γ2, and is thus insensitive to the stored quantum information. b, Schematic of the charge sensing readout. A nearby QD (CS) is capacitively coupled to QDR but isolated by a high tunnel barrier that prevents direct tunnelling between them. A lead with a resonator is coupled to the CS for fast single-lead RF reflectometry34. c, Principle of charge sensing: changes in the charge on QDR alters the local electrostatic environment, shifting the CS resonance and modulating the reflected signal \(\varDelta {S}_{{\rm{R}}}^{11}\). d, Charge-stability diagram of QDR and CS measured through \(\varDelta {S}_{{\rm{R}}}^{11}\), with QDL held on resonance (VLD = 263 mV). To correct for cross-capacitance, the voltage on VCS is adjusted during VRD sweeps, resulting in an effective compensated axis \({V}_{{\rm{CS}}}^{{\rm{v}}}\) that is a linear combination of VRD and VCS. e, Charge-stability diagram of QDL and QDR with \({V}_{{\rm{CS}}}^{{\rm{v}}}\) fixed (grey mark in panel d). Outside the QDL resonance, QDR shows fixed charge when its energy is above the hybrid gap EABS. Below this gap, parity switching is visible as signal noise. Near the QDL resonance (VLD = 263 mV), the region of visible switching is reduced when QDL becomes lower in energy than QDR. f–h, The signal-to-noise ratio (SNRR, panel f), parity polarization (PR, panel g) and average switching time (τavg, panel h) obtained from time traces measured in the dotted grey box of panel e. In these panels, the grey contours indicate where SNRR = 0.5. The SNRR vanishes at the centre of the diagram (in which VLD ≈ 262.5 mV), indicating that the CS is insensitive to the parity states at this point. This dataset was acquired simultaneously with the quantum capacitance measurements in Fig. 3e and used to compute correlations in Fig. 4.
Extended Data Fig. 8 Time traces with weakly coupled QDs.
a–d, Charge-stability diagram measured by quantum capacitance (panel a) in a weak interdot coupling regime (VH = 1.445 V). The dotted grey box marks the region in which time traces were measured to estimate the signal-to-noise ratio (SNRM, panel b), parity polarization (PM, panel c) and average switching time (τavg, panel d). We infer the weak coupling from the presence of signal when a QD is aligned to the lowest-energy state in the hybrid, as highlighted by the arrows in panels a and b and discussed in Fig. 2. In these panels, the grey contours indicate where SNRM = 0.5. The probing frequency for the middle resonator was 341 MHz instead of 340.8 MHz, as in the rest of the text. The time traces in this figure were averaged in time bins of 200 μs (instead of 150 μs) to ensure that the averaging time is an exact multiple of inverse of the sampling rate (10 kHz). e–h, Same measurements by charge sensing. Charge-stability diagram of QDL and QDR (panel e). Outside the QDL resonance, QDR shows fixed charge when its energy is above the hybrid gap EABS. Below this gap, parity switching is visible as signal noise. Near the QDL resonance, the region of visible switching is reduced when QDL becomes lower in energy than QDR. The region in which even and odd states are indistinguishable is narrower than in Extended Data Figs. 7 and 9, consistent with small coupling between QDs (see Methods section ‘Model’). i, Pearson correlation coefficient between time traces measured with quantum capacitance and charge sensing. Similarly to Extended Data Fig. 5q, the transition region between positive and negative correlation (black arrows) is tilted. Furthermore, this transition is more sharply defined, in line with the low coupling (see Methods section ‘Model’).
Extended Data Fig. 9 PMM sweet spot reproduction with different QD resonances.
a–d, Charge-stability diagram at VH = 1.625 V measured by quantum capacitance (panel a), with the corresponding signal-to-noise ratio (SNRM, panel b), parity polarization (PM, panel c) and average switching time (τavg, panel d) for different QD resonances in which t ≈ Δ. Panels b–d were measured inside the dotted grey dotted box of panel a. In these panels, the grey contours indicate where SNRM = 0.5. e–h, Same measurements by means of charge sensing. The charge-stability diagram of QDL and QDR is shown in panel e. Outside the QDL resonance, QDR shows fixed charge when its energy is above the hybrid gap EABS. Below this gap, parity switching is visible as signal noise. Near the QDL resonance (VLD = 286.8 mV), the region of visible switching is reduced when QDL becomes lower in energy than QDR. The signal-to-noise ratio (SNRR, panel f), parity polarization (PR, panel g) and average switching time (τavg, panel h) are obtained from time traces measured in the dotted grey box of panel e. In these panels, the grey contours indicate where SNRR = 0.5. The region in which SNRR vanishes near the centre of the diagram is wider than for Extended Data Figs. 7 and 8, indicating a stronger coupling. i, Pearson correlation coefficient between time traces measured with quantum capacitance and charge sensing. The black arrows indicate the region in which the CS is insensitive to parity-switching events, resulting in vanishing correlation between time traces measured with quantum capacitance and charge sensing. As for Fig. 4, this line is horizontal (μLD = 0), which occurs only at the PMM sweet spot (t = Δ). j–l, Examples of uncorrelated (j), negatively correlated (k) and positively correlated (l) time traces. The marks in panel i indicate the specific gate values at which these time traces were measured. Note that, for panels k and l, the reflected signal is, respectively, positively and negatively correlated, opposite to the state assignment (see Methods section ‘State assignment’ for more details).
Extended Data Fig. 10 Theory model of quantum capacitance, charge sensing and quasiparticle poisoning dynamics.
a,b, Quantum capacitance (a) and right QD charge difference (b) for tL = tR = 0.3EABS, θ = π/8 and μC = 0.75ΔP, corresponding to a PMM sweet spot. Red and blue areas indicate whether the odd or even states yield the largest signal and white for which the states are indistinguishable. c–e, Zoom-ins of QD charge difference with tL = tR = 0.5EABS, θ = π/8 and μC = 0.75ΔP (c), μC = 2.5ΔP (d) and μC = 0.3ΔP (e). Here various tilts of the \({\mu }_{{\rm{LD}}}=\frac{\varDelta -t}{\varDelta +t}{\mu }_{{\rm{RD}}}\) line are shown as emphasized by the black arrows. Grey arrows indicate the width of the transitions, set by Δ and t. f–h, Parity polarization maps for tL = tR = 0.4EABS, θ = π/8, Tp = 0.2ΔP, with μC = 0.75ΔP (f), μC = 2.5ΔP (g) and μC = 0.3ΔP (h). These maps match the experimental behaviour observed in Extended Data Fig. 5. i, Average parity lifetime τavg = 1/(Γe→o + Γo→e) for same parameters as in panel f. j, Schematic of the poisoning process, ΓΔ, which decreases the lifetime in the centre of i by means of poisoning to the excited states. The x-axis corresponds to the blue arrow in i. Details on the model can be found in Methods section ‘Model’, with calculations on the quantum capacitance and charge sensing signals in subsection ‘Quantum capacitance and charge sensing’ and state occupation in subsection ‘Occupation’.
Extended Data Fig. 11 Quantum capacitance parity readout of two minimal chains in a qubit geometry.
a, Schematic of a device with two minimal Kitaev chains coupled to a common superconductor and an off-chip resonator. Each chain contributes a parity-dependent quantum capacitance: \({C}_{{\rm{Q}}}^{1(2)}\propto 1/{\varDelta }_{1(2)}\) (if the chain is even) and \({C}_{{\rm{Q}}}^{1(2)}=0\) (if the chain is odd), in which Δ1(2) is the CAR coupling of the first (second) chain. The total quantum capacitance for the four possible parity states is \({C}_{{\rm{Q}}}^{{\rm{e}}{\rm{e}}}\propto 1/{\Delta }_{1}+1/{\Delta }_{2}\), \({C}_{{\rm{Q}}}^{{\rm{e}}{\rm{o}}}\propto 1/{\Delta }_{1}\), \({C}_{{\rm{Q}}}^{{\rm{o}}{\rm{e}}}\propto 1/{\varDelta }_{2}\) and \({C}_{{\rm{Q}}}^{{\rm{oo}}}=0\). If the chains are asymmetric (Δ1 ≠ Δ2), four states can be distinguished. On the other hand, if Δ1 ≈ Δ2, the two globally odd states cannot be distinguished. Nevertheless, the two even states can always be distinguished both from each other and from the globally odd state41. b, False-coloured scanning electron micrograph of a device with two chains. The nanowire is depleted between inner dots QDR1 and QDL2 to prevent inter-chain tunnelling. Ohmic contacts (yellow) were added in a second fabrication step. The rightmost QD was shifted by one gate to the right, because the gate closest to the superconductor was too strongly screened to form a suitable tunnel barrier. c,d, Charge-stability diagram of the left chain while the QDs of the right chain are off resonance (c) and vice versa (d). In this case, the quantum capacitance readout is sensitive to the parity of only one chain. Parity-switching events appear as telegraph noise in the centre of the charge-stability diagrams, as discussed in Fig. 2. For this measurement and the following ones, the two chains were not fine-tuned to the sweet spot t = Δ. e,f, Charge-stability diagram of the left chain while the QDs of the right chain are on resonance (e) and vice versa (f). The telegraph noise in the background corresponds to parity-switching events of the chain whose QDs are not being detuned. At the centre of the charge-stability diagram, all of the QDs are on resonance and more than two states can be detected. g, Time trace with the QDs of both chains on resonance, taken at the gate settings marked by grey crosses in c–f. Three levels are resolved, consistent with Δ1 ≈ Δ2, at which the odd states merge. A Gaussian mixture model (see Methods section ‘Time traces measurement and analysis’) identifies the states, with the central state twice as probable. This result demonstrates single-shot readout in the computational basis |ee⟩, |oo⟩ for a potential PMM qubit, while also enabling detection of leakage into the global odd manifold. A full investigation of parity–qubit operation will be the subject of future work. h, Histogram of the complex time trace plotted in g. The one-dimensional time trace was obtained by projecting the complex data onto the principal component axis, drawn here with a dotted line. i, Distribution of the dwell times for the three identified states. The good match with an exponential model confirms that the switching processes are Poissonian. The average dwell times are comparable with those measured in the single-chain device. j, Transition rates and their standard deviation for the three-state switching process. No direct |ee⟩ ↔ |oo⟩ were observed. The corresponding matrix entries are the estimated upper bound (95% confidence level). This confirms the absence of inter-chain tunnelling.
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van Loo, N., Zatelli, F., Steffensen, G.O. et al. Single-shot parity readout of a minimal Kitaev chain. Nature 650, 334–339 (2026). https://doi.org/10.1038/s41586-025-09927-7
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DOI: https://doi.org/10.1038/s41586-025-09927-7
