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.