Abstract

The fusion of non-Abelian anyons is a fundamental operation in measurement-only topological quantum computation¹. In one-dimensional topological superconductors (1DTSs)^2–4, fusion amounts to a determination of the shared fermion parity of Majorana zero modes (MZMs). Here we introduce a device architecture⁵ that is compatible with future tests of fusion rules. We implement a single-shot interferometric measurement of fermion parity^6–11 in indium arsenide–aluminium heterostructures with a gate-defined superconducting nanowire^12–14. The interferometer is formed by tunnel-coupling the proximitized nanowire to quantum dots. The nanowire causes a state-dependent shift of the quantum capacitance of these quantum dots of up to 1 fF. Our quantum-capacitance measurements show fluxh/2e-periodic bimodality with a signal-to-noise ratio (SNR) of 1 in 3.6 μs at optimal flux values. From the time traces of the quantum-capacitance measurements, we extract a dwell time in the two associated states that is longer than 1 ms at in-plane magnetic fields of approximately 2 T. We discuss the interpretation of our measurements in terms of both topologically trivial and non-trivial origins. The large capacitance shift and long poisoning time enable a parity measurement with an assignment error probability of 1%.

A device architecture based on indium arsenide–aluminium heterostructures with a gate-defined superconducting nanowire allows single-shot interferometric measurement of fermion parity and demonstrates an assignment error probability of 1%.

Main

To make use of a topological phase for quantum computation, it is crucial to manipulate and measure the topological charge. This can be achieved through protected operations such as braiding and fusing non-Abelian anyons, which offer exponential suppression of errors induced by local noise sources and a discrete set of native operations^15–17. Protocols for measurement-only topological quantum computation simplify these operations, reducing them to fusion alone^1,5. This fundamental measurement is sufficient to enact all topologically protected operations. New error-correction schemes have been developed to take advantage of these operations^18–20. The robustness against errors and simplicity of control offered by this approach make measurement-based topological qubits a promising path towards utility-scale quantum computation, in which managing the interactions of millions of qubits is necessary^21–24.

1DTSs^2–4 are a promising platform for building topological qubits. Quantum information is stored in the fermion parity of MZMs localized at the ends of superconducting wires and projective measurements of the fermion parity are used to process quantum information and perform qubit-state readout^25,26. The fermion parity shared by a pair of MZMs can be determined through an interferometric measurement^3,6–9. Several conceptual designs for topological qubits incorporate such interferometers^5,10,11,27. These proposals require time-resolved measurements of the fermion parity in the interference loop, which cannot be accomplished with dc transport measurements of the time-averaged fermion parity²⁸.

In this paper, we demonstrate such a time-resolved measurement, thereby validating a necessary ingredient of topological quantum computation. The measurement technique is based on examining the quantum capacitanceC_Q of a quantum dot coupled to the nanowire^5,29–31 (Fig.1) and allows determination of the parity in a single shot. We achieve an assignment error probability of 1% for optimal measurement time. By itself, this measurement does not unequivocally distinguish between MZMs in the topological phase and fine-tuned low-energy Andreev bound states in the trivial phase^32–40 but it does require the low-energy state to be supported at both ends of the wire and very weakly coupled to other low-energy fermionic states. Moreover, it provides a measurement of the state’s energy with single-μeV resolution. These features of the measurement strongly constrain the nature of the low-energy state.

Device design and setup

We introduce a device architecture enabling projective measurements of fermion parity^{5,10,11,27,29,41,42}. The device comprises two primary components, as illustrated in Fig.1. The first component is a nanowire that will have MZMs at its ends if it is in a 1DTS state. The second component consists of quantum dots, which are designed to couple pairs of MZMs in an interferometric loop.

The nanowire in this device is based on a gated superconductor–semiconductor heterostructure and defined by a narrow Al strip that suppresses depletion underneath it^12–14. Device fabrication and details of the heterostructure design are discussed in Sections 1.2 and 1.3 of theSupplementary Information, respectively. The Al strip is grounded and continuous throughout, but there are separate ‘plunger’ gates that define five sections of the wire. One of them is shown schematically in Fig.1c and all five are visible in the scanning electron microscopy image in Fig.2b. Although there are no breaks in the Al, the plunger gates independently control the density in each section. (See Supplementary Fig.1 and Section 1.1 of theSupplementary Information for a complete device schematic and gate-naming convention; throughout the paper,V_i refers to the dc voltage applied to gatei.) A topological qubit would require tuning the second and fourth segments, each of lengthL ≈ 3 μm, into the 1DTS state, whereas the other three would be fully depleted underneath the Al strip (see Supplementary Fig.1 for details). Here we focus on the second section shown in Fig.1c and implement a parity measurement using its associated interferometer.

Our readout circuit is based on dispersive gate sensing of a triple quantum dot interferometer (TQDI): three electrostatically defined quantum dots that, together with the second nanowire section, form a loop threaded by a flux,Φ (Fig.1a,c). We controlΦ by varying the out-of-plane magnetic field,B_⊥. The TQDI has two smaller dots (dots 1 and 3), which serve as tunable couplers providing control over, respectively, the tunnel couplingst_L andt_R. The smaller dots are connected to the ends of the 1DTS through tunnel couplingst_mi, in whichi = 1, 2, and to the long quantum dot (dot 2) that connects to dot 1 and dot 3 through tunnel couplingst₁₂ andt₂₃, respectively. The quantum capacitance,C_Q, of dot 2 is read out through dispersive gate sensing using an off-chip resonator circuit in a reflectometry setup⁴³; a detailed description is given in Section 1.4 of theSupplementary Information.

We have developed an rf-based quantum dot–MZM tuning protocol that we use to balance the arms of the interferometer. We measureC_Q in a configuration in which one of the small dots is maximally detuned, effectively interrupting the loop. Comparing these measurements with simulations, we extract the couplingst₁₂, t₂₃, t_m1 andt_m2 (see Section 2.5 of theSupplementary Information). This measurement protocol expands on the dc transport techniques proposed in refs. ^44,45 and demonstrated in ref. ⁴⁶. Our rf-based protocol offers μeV-level resolution for coupling extraction, which enables tuning the effective dot-to-wire couplingst_L andt_R. Once we have determined the appropriate voltages for quantum dots 1 and 3, we proceed with interferometry measurements. Section 4 of theSupplementary Information contains further details of the tune-up procedure.

Fermion parity measurement and interpretation

To measure a time record of the fermion parity, we tune up the TQDI and perform a sequence of nearly 1.5 × 10⁴ consecutive measurements of the resonator response, each with an integration time of 4.5 μs, thereby recording a time trace of total length 67 ms. To improve visibility and compare with theoretical predictions, we downsample the time trace by averaging over a 90-μs window. By comparing the measured resonator response with a reference trace (taken with dot 2 in a Coulomb valley), we convert it to a${\tilde{C}}_{\mathrm{Q}}$ record, which includes a field-dependent shift ofC_Q that cancels out of ΔC_Q (see equation (28) in theSupplementary Information).

We sweepV_QD2 to find charge transitions in dot 2 and, because the normal to the plane of the device is only slightly offset (<1°) from thex axis of the magnet, we sweep thex component of the magnetic fieldB_x in steps of 0.14 mT to study the dependence onΦ. OurB_x sweep range is offset from 0 so thatB_⊥ (which contains a contribution fromB_z) is swept symmetrically around 0. We use the topological gap protocol (TGP)¹⁴ to select an in-plane fieldB_∥ and a wire plunger gate voltageV_WP1 range (indicated, respectively, in Fig.1a,c) for our measurements, as discussed in Section 4 of theSupplementary Information. The readout system parameters that we achieve are not strongly dependent on these values. For measurement A1 of device A, the relevant regime isB_∥ ≈ 1.8 T andV_WP1 ≈ −1.832 V.

For appropriately tuned quantum dot plungers, in particular forV_QD2 close to resonance, the measured${\tilde{C}}_{\mathrm{Q}}$ record exhibits switches between two capacitance values that differ by a ΔC_Q(B_x) that oscillates as a function ofB_x. At someB_x, there are no visible switches, as in Fig.3a, so ΔC_Q(B_x) vanishes. At genericB_x, there is a clear random telegraph signal (RTS), which is shown in Fig.3d for theB_x that corresponds to maximal ΔC_Q(B_x). From a histogram of all${\tilde{C}}_{\mathrm{Q}}$ observed within this time trace, we extract an achieved SNR of 5.01 in 90 μs (Fig.3e,f) or, equivalently, an SNR of 1 in 3.6 μs (see Section 3.3 of theSupplementary Information). As demonstrated in Fig.3g, the intervals between switches follow an exponential distribution with a characteristic timeτ_RTS ≈ 2 ms. By plotting histograms of the${\tilde{C}}_{\mathrm{Q}}$ time traces as a function ofB_x, as shown in Fig.3h, we observe aB_x-dependent bimodal distribution of${\tilde{C}}_{\mathrm{Q}}$ values with peaks separated by ΔC_Q(B_x). The oscillation period of ΔC_Q(B_x) is 1.9 ± 0.1 mT, which is consistent with the expected flux ofh/2e through the interference loop in this device geometry. We interpret the RTS inC_Q as originating from switches of the fermion parity in the wire; see Section 7.3 of theSupplementary Information for details.

The visibility and phase of the oscillations vary between successive charge transitions in dot 2. We illustrate this by showing the kurtosisK(C_Q) (which detects bimodality; see Section 3.2 of theSupplementary Information) of the${\tilde{C}}_{\mathrm{Q}}$ time traces for several different charge transitions in Fig.3i. A similar difference in the visibility of flux-induced oscillations across different charge transitions was recently observed in a double quantum dot interferometer experiment⁴⁷. In Section 6 of theSupplementary Information, we discuss oscillations with different periods that are observed at other points in the parameter space of the device.

We support this interpretation by reproducing our results with quantum dynamics simulations that incorporate rf drive power, charge noise and temperature. To build intuition for those simulations, we use an idealized model (see Section 2.2 of theSupplementary Information) subject to the following assumptions (which we will later relax): the wire is in the topological phase and there are no sub-gap states other than the MZMs; the charging energy and level spacing in the dots are much greater than the temperature; dots 1 and 3 are sufficiently detuned that their influence is fully encapsulated in the effective couplingst_L andt_R to MZMs at the ends of the wire (see Fig.1a); and the drive frequency and power are both negligible. In this limit, the quantum capacitance as a function of the total fermion parity in the quantum dot–wire system,Z, is given by

in whichE_D is the detuning from the charge-degeneracy point,α is the lever arm of the plunger gate to the dot,E_M is the MZM energy splitting andT is the temperature. The net effective tunnelling that results from the interference between different trajectories from the dot to the MZMs and back,t_C(Z, ϕ), is

Hereϕ is the phase difference betweent_L andt_R, which is controlled by the magnetic fluxΦ through the interference loop created by the dot, the wire and the tunnelling paths between them according toϕ = 2πΦ/Φ₀ + ϕ₀, in whichΦ₀ = h/e andϕ₀ is a flux-independent offset. To capture the extent to whichC_Q can be used to discriminate betweenZ = ±1, it is convenient to introduce

The interferometer must be well balancedt_L ≈ t_R for ΔC_Q to be large according to equation (1). WhenE_M = 0, ΔC_Q exhibits maxima along theE_D = 0 line, with flux periodicityh/2e.

For detailed comparison with experiments, we use the methods discussed in Sections 2.4 and 2.5 of theSupplementary Information to simulate a more complete model of the device and readout chain that includes the full triple-dot system, incoherent coupling to the environment (using parameters inferred from separate measurements; see Sections 9 and 10 of theSupplementary Information) and measurement backaction. Crucially, this approach allows us to incorporate different noise sources in a systematic and quantitative way without any free parameters. The simulated dynamicalC_Q, defined in Section 2.3 of theSupplementary Information, is shown in Fig.4. TheC_Q histograms in Fig.4a reveal twoh/e-periodic branches (one shown in red and the other in blue), associated with the two parities of the coupled system. If the fermion parityZ were perfectly conserved, then the device would remain in one of the two parity eigenstates and theΦ dependence would follow either the blue or the red trace in Fig.4a. However,Z should fluctuate on a timescale given by the quasiparticle poisoning timeτ_qpp. Hence, in traces over times longer thanτ_qpp, a bimodal distribution ofC_Q values is expected, that is, both the blue and red traces in Fig.4a. Consequently, the kurtosisK(C_Q) exhibits minima at which ΔC_Q is peaked, as shown in Fig.4b, and time traces taken at these points will exhibit a telegraph signal composed of switches between the valuesC_Q(1, ϕ) andC_Q(−1, ϕ). Comparing Fig.4 with Fig.3h,i, we find good overall agreement of both the histograms and the kurtosis. We find a maximum ΔC_Q(Φ) ≈ 1 fF, which is consistent with our measurements in Fig.3. This agreement extends to other parameter regimes, such as when the interferometer is poorly balanced or the splittingE_M is sizeable, as discussed in Section 6 of theSupplementary Information.

A second measurement of device A and a measurement of a second device (device B) give results in qualitative agreement with those of measurement A1, demonstrating the reproducibility of the observed phenomena (Section 5 of theSupplementary Information). We have tested our interpretation by: (1) disconnecting the dots from the wire; (2) measuring at fields of 0.8 T below the region identified by TGP; (3) intentionally injecting quasiparticles into the superconductor and observing the effect onτ_RTS; and (4) comparing the quasiparticle density measured in a separate test structure with that inferred according to the hypothesis thatτ_qpp = τ_RTS ≈ 2 ms (Section 7 of theSupplementary Information).

By extending the model introduced above, we have analysed the quasi-MZM scenario discussed in previous works^37–39,48. We introduce an extra pair of ‘hidden’ Majorana modes that are weakly coupled to each other and to the visible MZMs, which themselves are coupled to quantum dots 1 and 3. Together, the hidden and visible MZMs form a trivial low-energy state at each end of the wire. This scenario can occur in the trivial phase, in which it requires some fine-tuning to make the couplings small. In Section 2.7 of theSupplementary Information, we show that the hidden Majorana modes suppress ΔC_Q owing to fast fermion tunnelling between them and the visible MZMs. This effect completely washes out the flux-dependent bimodality unless the coupling between the ‘hidden’ Majorana modes and the visible MZMs is less than 1 neV or the hidden Majorana modes are effectively gapped out, as shown in Supplementary Fig.4.

Discussion and outlook

We have presented dispersive gate-sensing measurements of the quantum capacitance in InAs–Al hybrid devices using a system architecture that can be adapted to other materials platforms^49,50. After tuning the nanowire density and in-plane magnetic field into the parameter regime identified by the TGP¹⁴ and balancing the interferometer formed by the nanowire and the quantum dots, we observed a flux-dependent bimodal RTS in the quantum capacitance, which we interpret as switches of the parity of a fermionic state in the wire. We have fit these data to a model in which the fermion parity is associated with two MZMs localized at the opposite ends of a 1DTS and find good agreement. These measurements do not, by themselves, determine whether the low-energy states detected by interferometry are topological. However, our data tightly constrain the allowable energy splittings in models of trivial Andreev states.

In conclusion, our findings represent substantial progress towards the realization of a topological qubit based on measurement-only operations. Single-shot fermion parity measurements are a key requirement for a Majorana-based topological quantum computation architecture.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41586-024-08445-2.

Supplementary information

Acknowledgements

We thank H. Beidenkopf, S. Das Sarma, L. Glazman, B. Halperin, A. Kou, K. Moler, W. Pfaff and M. Rudner for discussions. We thank E. Lee and T. Ingalls for assistance with the figures. We are grateful for the contributions of A. Dokania, A. Efimovskaya, L. Johansson and A. Mullally at an early stage of this project. We have benefited from interactions with P. Accisano, P. Bonderson, J. Borovsky, T. Brown, G. Campbell, S. Chakravarthi, K. Das, N. Dick, R. Gatta, H. Gavranovic, M. Goulding, J. Knoblauch, S. Jablonski, S. Kimes, J. Kuesel, J. Mattinson, A. Moini, T. Noonan, D. O. Fernandez Pons, L. Sanderson, M. P. da Silva, P. Strøm-Hansen, S. Suzuki, M. Turner, R. Yu and A. Zimmerman.

Author contributions

The Microsoft Azure Quantum team conceived the technology reported in this article and designed, fabricated and operated the device and system reported here. All authors wrote and revised the manuscript and the Supplementary Information.

Peer review

Data availability

The datasets associated with the figures in this paper are available at Zenodo⁵¹ (10.5281/zenodo.14804379). Further data from devices A and B demonstrating the functionality of this device architecture for fermion parity measurements (namely, quantum dot charging energies and level spacings, inter-dot couplings, dot–wire couplings and wire plunger gates) are available from the corresponding author on request.

Code availability

The source code that performs the analysis and generates the figures in this paper are available at our public GitHub repository at github.com/microsoft/azure-quantum-parity-readout.

Competing interests

The authors declare no competing interests.

Footnotes

Supplementary information

The online version contains supplementary material available at 10.1038/s41586-024-08445-2.

References

Associated Data

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