Procedure of tuning up a three-site artificial Kitaev chain based on transmon measurements

Bingbing Tong; Bing Li; Duolin Wang; Enna Zhuo; Fanming Qu; Guangtong Liu; Jie Shen; Li Lu; Peiling Li; Xiang Wang

REVIEW 4 major objections 4 minor 1 cited by

Procedure of tuning up a three-site artificial Kitaev chain based on transmon measurements

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A three-site artificial Kitaev chain has three types of sweet spots—ECT-dominated, genuine, and CAR-dominated—and the two off-balance types retain a finite excitation gap at half a flux quantum, which makes them the practical operating…

desk verdict A useful tuning recipe with a correct classification; the transmon readout formula needs a validity check before the central convenience claim is accepted. read the letter →

arxiv 2505.15317 v1 pith:NKRXQJNY submitted 2025-05-21 cond-mat.mes-hall quant-ph

Xiaozhou Yang , Zhaozheng Lyu , Xiang Wang , Enna Zhuo , Yunxiao Zhang , Duolin Wang , Yukun Shi , Yuyang Huang

show 9 more authors

Bing Li Xiaohui Song Peiling Li Bingbing Tong Ziwei Dou Jie Shen Guangtong Liu Fanming Qu Li Lu

This is my paper · ORCID

classification cond-mat.mes-hallquant-ph

keywords artificialKitaevchainMajoranaboundstatestransmonreadoutsweetspotselasticcotunnelingcrossedAndreevreflectionparityquantumdot-superconductorarray

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper presents a step-by-step procedure for tuning a three-site artificial Kitaev chain—a quantum-dot–superconductor linear array—into its sweet spots, using the plasma-mode spectrum of an integrated transmon circuit as a non-destructive parity readout. It establishes that the sweet spots are not one condition but three classes, depending on whether elastic cotunneling (ECT), crossed Andreev reflection (CAR), or neither dominates between neighboring dots. The ECT-dominated and CAR-dominated sweet spots keep a finite excitation gap at phase difference $\phi = \pi$ even though the odd- and even-parity ground states remain degenerate, while the genuine sweet spot's gap collapses there; that finite gap suppresses level crossings and Landau-Zener losses during fast phase manipulations. A sympathetic reader would care because the paper turns Majorana-bound-state tuning from a search over many parameters into a guided recipe with predicted transmon signatures at every step.

What carries the argument

The load-bearing relation is Eq. (7), $\delta E_p = (\hbar \omega_p - E_C) + \frac{1}{2\lambda^2}\partial_\phi^2 E_{\mathrm{AKC}}|_{\phi=\phi_{\mathrm{ext}}}$, obtained by treating the chain as a perturbation on the transmon and keeping the three lowest unperturbed transmon states, dropping transitions between chain eigenstates, and working to order $\lambda^{-2}$. It turns the measurable transmon plasma-mode spacing into a direct probe of the curvature of the chain ground state with respect to the superconducting phase, so the sign and discontinuity of $\partial_\phi^2 E_0$ encode parity, and dips in it encode sweet spots. The second structural object is the Majorana-operator rewriting of the three-site Kitaev Hamiltonian, Eq. (9), which shows that at $|t|=|\delta|$ one pair of end Majorana operators decouples, while for unequal $|t|$ and $|\delta|$ three Majoranas pair into a conventional fermion and leave one uncoupled end mode; this explains why degenerate ground states survive away from exact balance.

What would settle it

A decisive check: take a real three-site device, sweep the external flux, and record the transmon plasma-mode spacing as a function of the on-site energies; if the sign change and discontinuity of $\partial_\phi^2 E_0$ at the odd-even parity boundary does not appear at the predicted gate voltages, or if the extracted $\varepsilon_{2,\mathrm{ex}}$ deviates from zero by more than about $0.02\Delta$ at weak coupling, then Eq. (7) is not a faithful map from transmon spectrum to chain ground-state curvature. A second check is excited-state spectroscopy near $\phi = \pi$: a genuine sweet spot should drive the excitation gap to zero, whereas ECT- and CAR-dominated sweet spots should leave it finite.

Watch

Extended reading notes

Core claim

The central claim is that in a fully coupled quantum dot–superconductor–quantum dot–superconductor–quantum dot chain, sweet spots—parameter regions where $\delta E_{00}^{oe} \approx 0$ and the Majorana polarization at the ends $|\mathrm{MP}_1| > 0.95$—persist across a range of ECT/CAR imbalance, not only at exact balance. The paper classifies them as ECT-dominated ($|t| > |\delta|$), genuine ($|t| = |\delta|$), and CAR-dominated ($|t| < |\delta|$) sweet spots. At $\phi = \pi$ the excitation gap $\delta E_{\mathrm{gap}}$ vanishes only at the genuine sweet spot and remains finite at the other two, even though the ground-state degeneracy persists at all three. Because a finite gap avoids energy-level crossings between plasma modes and chain excitations and suppresses Landau-Zener transitions, the paper concludes the ECT- and CAR-dominated sweet spots are the more convenient operating points for transmon-based measurements, and it provides iterative two-parameter tuning steps to reach a sign-ordered three-site chain.

Load-bearing premise

The recipe assumes that the shift in the transmon's two lowest energy levels is dominated by the curvature of the chain's ground-state energy, with higher transmon levels, transitions between chain states, and higher-order corrections making no practical difference; if those contributions are significant, the predicted parity and sweet-spot signatures would not appear as calculated.

Editorial extensions

If this is right

Tuners can aim for an ECT- or CAR-dominated sweet spot instead of exact ECT/CAR balance, since those keep a finite excitation gap at $\phi = \pi$ while preserving degenerate ground states.
The sign of $\partial_\phi^2 E_0$ provides a non-destructive parity readout: negative curvature marks an odd-parity ground state, positive marks even, with a discontinuity at the boundary.
In the QD-S-QD-S configuration, $\varepsilon_4$ barely shifts the sweet-spot value of $\varepsilon_1$, so alternating two-parameter adjustments between the two four-dot configurations converge to a sign-ordered three-site chain.
At the genuine sweet spot the odd/even curvature splitting is confined to a narrow window near $\phi = \pi$, while at the CAR-dominated sweet spot it is broader but smaller, giving an explicit experimental trade-off between distinguishability and phase-control precision.
Two coincident transmon responses for the degenerate odd and even ground states at a sweet spot would be a direct non-destructive signature of Majorana physics, a signature the authors note has not yet been reported.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One extension beyond the paper: the same classification should carry over to longer parity-odd chains, so the practical operating point for scalable readout may be a deliberately imbalanced ECT/CAR ratio rather than exact balance.
A testable follow-up is to sweep $\varepsilon_2$ and $\varepsilon_4$ across zero while monitoring the transmon curvature; the predicted transition from CAR dominance to ECT dominance should appear as a crossing in the extracted coupling imbalance at the genuine sweet spot.
Because the transmon readout is fast and non-destructive, the same curvature signature could be used to measure the chain's parity lifetime or to run automated feedback tuning, which the paper does not analyze.
If finite excitation gaps at imbalanced sweet spots persist under quasi-static flux noise, those points may also be the better choice for coherence of a Majorana qubit, though the paper does not address decoherence.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read the paper. It's a numerical recipe paper, not a discovery paper, but the recipe is sensible and the classification is a clean reframing. The new piece is the distinction between ECT-dominated, genuine, and CAR-dominated sweet spots in a three-site chain, and the observation that the non-genuine types retain a finite excitation gap at phi = pi, which helps avoid Landau-Zener and level-crossing issues in transmon readout. That's a solid and useful point. The iterative two-configuration tuning protocol (Step 2) is also new and seems experimentally practical.

The paper does well in grounding the classification in the Kitaev model (Eq. 9, Fig. 4) and then showing the AKC model reproduces the same features (Fig. 5). The numerics look internally consistent, and the Majorana polarization/wavefunction checks support the identification. No circular fitting—everything is computed from the same Hamiltonian, which is the right approach for a protocol proposal.

Soft spots: Eq. (7) is the bridge between the transmon measurement and the chain curvature. The derivation is only in the Supplementary Material, and the main text doesn't state the validity conditions. The stress-test concern is real: the finite gap at the non-genuine sweet spots in the Kitaev-model illustration is about 0.03Δ; if Δ is a few tens of GHz and the transmon frequency is several GHz, the off-resonant assumption behind second-order perturbation theory may not hold. The authors should either provide a combined transmon-chain spectrum for the operating parameters or at least report hbar*omega_p/Δ and the actual delta_E_gap values. Without that, the 'more conveniently accessed' claim is conditional.

Also, the parameters are borrowed without sensitivity analysis. That's minor—the qualitative features are probably robust—but it would be good to see a scan over E_Z and t_d.

Bottom line: this is a useful paper for experimental groups working on quantum-dot-based Majorana chains and transmon readout. It deserves peer review; a serious referee should ask for the Eq. (7) derivation to be moved or summarized, for the validity check described above, and for a parameter-robustness statement.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a step-by-step procedure for tuning a three-site artificial Kitaev chain into sweet spots using the plasma spectra of an integrated transmon circuit. The microscopic model couples three quantum dots through two superconducting dots, with the transmon phase controlled by external flux. The central claim is that sweet spots can be classified into ECT-dominated, genuine, and CAR-dominated types according to the relative strengths of elastic cotunneling and crossed Andreev reflection, and that the non-genuine types retain a finite excitation gap at phase phi = pi, making them more convenient for transmon-based readout. The claims are supported by exact diagonalization of the chain Hamiltonian and by a three-site Kitaev-model analogy.

Significance. If the claims hold, the paper offers practical guidance for experiments, including a concrete tuning sequence and discriminating signatures of different sweet-spot types. The classification is a transparent consequence of the Kitaev model, and the exact-diagonalization results are internally consistent. The paper gives explicit parameter values and phase diagrams, which makes the proposal actionable. However, the load-bearing measurement-mapping step, Eq. (7), rests on an adiabatic/off-resonant approximation that is not quantitatively validated, and no sensitivity analysis is supplied for the borrowed parameters. The manuscript is a theory proposal; no experimental data or reproducible code are provided.

major comments (4)

[§II, Eq. (7)] The derivation of Eq. (7) is relegated to the Supplemental Materials and assumes that only the three lowest transmon states contribute, that transitions between chain eigenstates are negligible, and that terms up to lambda^{-2} are sufficient. These assumptions are load-bearing because Eq. (7) is the only link between the measured plasma splitting and the chain curvature partial_phi^2 E_AKC. The manuscript never states the relevant energy scales: the chain excitation gap delta_E_gap at the operating sweet spots and the transmon frequency hbar*omega_p. In the Kitaev-model illustration (Figs. 4(d-f), delta0 = 0.055 Delta), delta_E_gap at phi = pi equals sqrt(2)|t - delta| approximately 0.031 Delta, which is comparable to or smaller than a typical transmon frequency (e.g., about 5 GHz if Delta is about 50 GHz). In that regime, virtual chain excitations are not far off resonance, and the neglected transitions would produce avoided crossings and level shifts not captured by Eq. (7). Please provide an explicit coupled-spectrum check or a quantitative adiabaticity condition, including numerical values of hbar*omega_p/Delta and delta_E_gap for the AKC sweet spots in Fig. 5.
[§II, parameters after Eq. (1)] All numerical results use E_Z = 3 Delta, U = 5 Delta, t_d = 0.4 Delta, and t_sc/t_sf = 3 without any sensitivity analysis. Since the paper proposes a procedure that experiments should follow, the robustness of the signatures to realistic parameter spread should be demonstrated. Please show, for example, how the sweet-spot locations and the parity and sweet-spot signatures in Figs. 2-5 shift when E_Z and t_sc/t_sf are varied within experimentally plausible ranges.
[§IV, Fig. 5(b,c)] The identification of the 'genuine sweet spot' in the AKC model is not fully consistent with the stated definition. In Fig. 5(c), the ECT and CAR strengths are equal at delta_epsilon_2 approximately 0.06 Delta, whereas the green arrow marking the genuine sweet spot is placed at delta_epsilon_2 = 0. The text attributes the offset to finite Zeeman splitting, but this means the point labeled genuine in Fig. 5(b) is not the point of equal ECT/CAR strength. Please clarify whether the arrow should be moved to the equal-strength point or whether the classification uses a different criterion.
[§IV, Figs. 4(d-f) and 5(d-f)] The central claim that ECT- and CAR-dominated sweet spots are 'more conveniently' accessed relies on the finiteness of the excitation gap at phi = pi, but the relevant gap values are not quantified for the AKC model. Only the Kitaev illustration gives delta_E_gap approximately 0.031 Delta at phi = pi. Whether this gap suppresses Landau-Zener transitions and off-resonant coupling with the transmon depends on the sweep rate and on hbar*omega_p relative to the gap. Please provide concrete numbers or a parametric estimate for the AKC sweet spots, and state the criterion under which the finite gap is experimentally useful.

minor comments (4)

[§V] In the Conclusion, 'sweep spots' should be 'sweet spots'.
[§III, Fig. 2(g)] The text states that epsilon_3 is fixed at -2.5 Delta, while the caption of Fig. 2(g) says epsilon_3 = -0.25 Delta. Please reconcile this discrepancy.
[§II, Eq. (7)] The term (hbar*omega_p - E_C) and the prefactor 1/(2 lambda^2) in Eq. (7) need a brief dimensional or derivation comment in the main text, since the reader cannot verify the prefactor without consulting the Supplemental Materials.
[§III, Step 2] The special sweet-spot line epsilon_2 = epsilon_4 approximately -0.694 Delta is assumed without derivation. A short justification, or a citation to where it is derived, would improve the self-containedness of the tuning protocol.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: transmon signatures are model outputs from the same microscopic Hamiltonian, and the only self-citation is motivational.

full rationale

All load-bearing quantities are computed from the microscopic Hamiltonians (1) and (8) by exact diagonalization and second-order perturbation theory; nothing is fitted to the target classification. Eq. (7) maps the transmon plasma splitting to ∂φ^2 E_AKC with the constants stated, so the transmon 'signatures' are model outputs, not reverse-engineered from the desired sweet-spot types. The three-type classification follows from the Kitaev-model zero-energy condition at ε_f=0 (Eq. (9)) and from the Γ_ECT/Γ_CAR comparison in Fig. 5(c); the finite excitation gap at ECT/CAR-dominated sweet spots is an analytically derived consequence (δE_gap=√2|t−δ| at φ=π) rather than an input. The AKC parameters in Fig. 5(b) are chosen to realize these regimes, which is the tuning procedure itself, not an independent prediction requiring external confirmation. The only self-citation, Ref. [40], is used to note that an odd-even parity transition was observed in the authors' previous transmon experiment; it motivates feasibility but does not support any derivation, so it is minor and non-load-bearing. The skeptical concern about Eq. (7) (neglect of chain transitions) is a validity/approximation issue, not circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claims rest on a microscopic model with parameters borrowed from Refs. [19,31], a perturbation-theory bridge to the transmon (Eq. 7), and a symmetry assumption (left-right symmetry); no new physical entities are introduced, and no parameter is fitted to experimental data in the main text. The 'special sweet spot' chosen as the target of the iterative procedure is an explicit operating-point assumption.

free parameters (6)

Zeeman splitting E_Z = 3 Delta
Chosen from Refs. [19,31]; sets the spin polarization energy scale and affects all charge-stability and parity diagrams.
Coulomb repulsion U = 5 Delta
Chosen from Refs. [19,31]; controls the charging energy of the quantum dots and the parity boundaries.
Inter-dot coupling strength t_d = 0.4 Delta
Chosen from Refs. [19,31]; sets the hopping amplitude between neighboring sites in the AKC model.
Spin-conserving to spin-flipping ratio t_sc/t_sf = 3
Chosen from Refs. [19,31] with the normalization t_sc^2 + t_sf^2 = 1; it fixes the relative strength of spin-conserving and spin-flipping tunneling.
Reference junction and charging energies E_J, E_C = E_J >> E_C (transmon limit)
No numerical values are given; the analysis only requires the transmon limit and the ratio E_J/E_C that enters the perturbation parameter lambda.
Assumed special sweet-spot line epsilon_2 = epsilon_4 = -0.694 Delta
The paper assumes this particular dashed line is the target special sweet spot for the iterative procedure in Step 2; it is a hand-chosen operating point, not an output of a fit.

assumptions (6)

domain assumption The QD-S array is described by the microscopic Hamiltonian of Eq. (1) with the stated parameter values.
The entire analysis is carried out within this standard AKC model; no alternative model or ab initio justification is given.
domain assumption The transmon phase difference theta is small and phi is approximately phi_ext because the reference junction is much stiffer than the chain (E_J >> E_chain).
Used to write the transmon Hamiltonian as Eq. (6) and to identify partial_phi^2 E_0 as the measurable quantity.
domain assumption Second-order perturbation theory with the three lowest transmon states, neglecting transitions between chain eigenstates and retaining terms up to lambda^{-2}, yields Eq. (7).
This is the mathematical bridge between the chain's energy curvature and the transmon level spacing; its derivation is deferred to the Supplemental Materials.
domain assumption Left-right symmetry of the three-site chain is assumed in Sec. IV B when analyzing the AKC model near the genuine sweet spot.
The analysis sets epsilon_2 = epsilon_4 and considers deviations delta_epsilon_2 = delta_epsilon_4 < 0.2 Delta, neglecting asymmetric configurations.
domain assumption The effective spinless Kitaev chain of Eq. (8) captures the relevant physics, with the microscopic ECT and CAR amplitudes Gamma computed from the original model.
This mapping is used to interpret the AKC results through the textbook Kitaev-chain picture and to derive the gap behavior at the three sweet-spot types.
domain assumption Spin-down electrons dominate the chain's low-energy behavior in the strong Zeeman field, so spin-up contributions can be neglected.
Used in Fig. 5(c), which shows only |Gamma_down-down_ECT| and |Gamma_down-down_CAR|; the quantitative effect of the neglected spin-up sector is not estimated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Procedure of tuning up a three-site artificial Kitaev chain based on transmon measurements." pith.science (2026). https://pith.science/paper/NKRXQJNY

@misc{pith2026250515317,
  author       = {Pith},
  title        = {Pith review of: Procedure of tuning up a three-site artificial Kitaev chain based on transmon measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKRXQJNY}},
  note         = {Machine review of arXiv:2505.15317}
}

read the original abstract

Artificial Kitaev chains (AKCs), formed of quantum dot-superconductor linear arrays, provide a promising platform for hosting Majorana bound states (MBSs) and implementing topological quantum computing. The main challenges along this research direction would include the tuning up of AKCs for hosting MBSs and the readout of the parity of the chains. In this work, we present a step-by-step procedure for tuning up a three-site AKC to its sweet spots based on the spectra of a transmon circuit which is integrated with the chain for the purpose of reading out the parity of the chain. The signatures of the transmon's plasma modes in each step, particular those related to the appearance of MBSs in the chain, will be given. We find that the sweet spots in a three-site AKC can be classified into three types based on the relative strengths of elastic cotunneling (ECT) and crossed Andreev reflection (CAR): ECT-dominated sweet spots, genuine sweet spots and CAR-dominated sweet spots. We show that the ECT-dominated and CAR-dominated sweet spots can be more conveniently accessed and utilized in transmon-based measurements.

Figures

Figures reproduced from arXiv: 2505.15317 by the authors.

**Figure 1.** Schematic illustration of a three-site AKC integrated into a transmon circuit with 𝐸୎ and 𝐸େ. The chain consists of three quantum dots (brown) and two superconducting dots (blue). The onsite energy 𝜀௜ and the inter-dot coupling 𝑡௜ can be tuned by the gate electrodes (yellow). The phase difference across the superconducting dots can be controlled by an external magnetic flux 𝜙ୣ୶୲. Ⅲ. Procedure Details In our model d… view at source ↗

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: (a) Schematic illustration of the QD-S-QD-S-QD configuration, i.e. a fully coupled threesite AKC. (b) 𝛿𝜀௝,ୗ୔ (𝑗 = 1,3,5) as functions of 𝛿𝜀(ଶ,ସ),ୗ୔ near the genuine sweet spot. The inset shows the |MP| at dot 1. At the genuine sweet spot, 𝜀ଵ,ୗ୔ = 𝜀ହ,ୗ୔ ≈ 2.915∆ , 𝜀ଶ,ୗ…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Entanglement dynamics in minimal Kitaev chains
quant-ph 2025-07 reject novelty 5.0 of 10

Two- and three-site Kitaev chains can dynamically generate maximally entangled two-qubit and GHZ-type three-qubit states, while a pure W state is forbidden by parity conservation.

Reference graph

Works this paper leans on

45 extracted references · 40 canonical work pages · cited by 1 Pith paper

[1]

A. Yu. Kitaev, Fault-Tolerant Quantum Computation by Anyons, Ann. Phys. 303, 2 (2003)

work page 2003
[2]

Nayak, S

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80, 1083 (2008)

work page 2008
[3]

S. D. Sarma, M. Freedman, and C. Nayak, Majorana Zero Modes and Topological Quantum Computation, Npj Quantum Inf. 1, 1 (2015)

work page 2015
[4]

A. Y . Kitaev, Unpaired Majorana Fermions in Quantum Wires, Phys.-Uspekhi 44, 131 (2001)

work page 2001
[5]

Fu and C

L. Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett. 100, 096407 (2008)

2008
[6]

Leijnse and K

M. Leijnse and K. Flensberg, Introduction to Topological Superconductivity and Majorana Fermions, Semicond. Sci. Technol. 27, 124003 (2012)

work page 2012
[7]

Sato and Y

M. Sato and Y . Ando, Topological Superconductors: A Review, Rep. Prog. Phys. 80, 076501 (2017)

work page 2017
[8]

R. M. Lutchyn, E. P. a. M. Bakkers, L. P . Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y . Oreg, Majorana Zero Modes in Superconductor–Semiconductor Heterostructures, Nat. Rev. Mater. 3, 52 (2018)

work page 2018

Show all 45 references

[9]

Flensberg, F

K. Flensberg, F. von Oppen, and A. Stern, Engineered Platforms for Topological Superconductivity and Majorana Zero Modes, Nat. Rev. Mater. 6, 944 (2021)

2021
[10]

Liu and H

W. Liu and H. Ding, Roadmap of the Iron-Based Superconductor Majorana Platform, Sci. China Phys. Mech. Astron. 66, 267002 (2023)

2023
[11]

J. D. Sau and S. D. Sarma, Realizing a Robust Practical Majorana Chain in a Quantum-Dot- Superconductor Linear Array, Nat. Commun. 3, 964 (2012)

2012
[12]

I. C. Fulga, A. Haim, A. R. Akhmerov, and Y . Oreg, Adaptive Tuning of Majorana Fermions in a Quantum Dot Chain, New J. Phys. 15, 045020 (2013)

2013
[13]

C.-X. Liu, G. Wang, T. Dvir, and M. Wimmer, Tunable Superconducting Coupling of Quantum Dots via Andreev Bound States in Semiconductor-Superconductor Nanowires, Phys. Rev. Lett. 129, 267701 (2022)

2022
[14]

Bordin, G

A. Bordin, G. Wang, C.-X. Liu, S. L. D. Ten Haaf, N. V an Loo, G. P. Mazur, D. Xu, D. Van Driel, F. Zatelli, S. Gazibegovic, et al., Tunable Crossed Andreev Reflection and Elastic Cotunneling in Hybrid Nanowires, Phys. Rev. X 13, 031031 (2023)

2023
[15]

Leijnse and K

M. Leijnse and K. Flensberg, Parity Qubits and Poor Man’s Majorana Bound States in Double Quantum Dots, Phys. Rev. B 86, 134528 (2012)

2012
[16]

Leumer, M

N. Leumer, M. Marganska, B. Muralidharan, and M. Grifoni, Exact Eigenvectors and Eigenvalues of the Finite Kitaev Chain and Its Topological Properties, J. Phys. Condens. Matter 32, 445502 (2020)

2020
[17]

Ezawa, Even-Odd Effect on Robustness of Majorana Edge States in Short Kitaev Chains, Phys

M. Ezawa, Even-Odd Effect on Robustness of Majorana Edge States in Short Kitaev Chains, Phys. Rev. B 109, L161404 (2024)

2024
[18]

Tsintzis, R

A. Tsintzis, R. S. Souto, and M. Leijnse, Creating and Detecting Poor Man’s Majorana Bound States in Interacting Quantum Dots, Phys. Rev. B 106, L201404 (2022)

2022
[19]

C.-X. Liu, S. Miles, A. Bordin, S. L. D. Ten Haaf, G. P. Mazur, A. M. Bozkurt, and M. Wimmer, Scaling up a Sign-Ordered Kitaev Chain without Magnetic Flux Control, Phys. Rev. Res. 7, L012045 (2025)

2025
[20]

R. A. Dourado, M. Leijnse, and R. S. Souto, Majorana Sweet Spots in 3-Site Kitaev Chains, arXiv:2502.19267
[21]

T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. D. ten Haaf, J.-Y . Wang, D. van Driel, F. Zatelli, et al., Realization of a Minimal Kitaev Chain in Coupled Quantum Dots, Nature 614, 445 (2023)

2023
[22]

Zatelli, D

F. Zatelli, D. van Driel, D. Xu, G. Wang, C.-X. Liu, A. Bordin, B. Roovers, G. P . Mazur, N. van Loo, J. C. Wolff, et al., Robust Poor Man’s Majorana Zero Modes Using Yu-Shiba-Rusinov States, Nat. Commun. 15, 7933 (2024)

2024
[23]

Bordin, C.-X

A. Bordin, C.-X. Liu, T. Dvir, F. Zatelli, S. L. D. ten Haaf, D. van Driel, G. Wang, N. van Loo, Y . Zhang, J. C. Wolff, et al., Enhanced Majorana Stability in a Three-Site Kitaev Chain, Nat. Nanotechnol. 1 (2025)

2025
[24]

S. L. D. ten Haaf, Q. Wang, A. M. Bozkurt, C.-X. Liu, I. Kulesh, P. Kim, D. Xiao, C. Thomas, M. J. Manfra, T. Dvir, et al., A Two-Site Kitaev Chain in a Two-Dimensional Electron Gas, Nature 630, 329 (2024)

2024
[25]

Kulesh, S

I. Kulesh, S. L. D. ten Haaf, Q. Wang, V . P. M. Sietses, Y . Zhang, S. R. Roelofs, C. G. Prosko, D. Xiao, C. Thomas, M. J. Manfra, et al., A Flux-Controlled Two-Site Kitaev Chain, arXiv:2501.15912

arXiv
[26]

S. L. D. ten Haaf, Y . Zhang, Q. Wang, A. Bordin, C.-X. Liu, I. Kulesh, V . P. M. Sietses, C. G. Prosko, D. Xiao, C. Thomas, et al., Observation of Edge and Bulk States in a Three-Site Kitaev Chain, Nature 1 (2025)

2025
[27]

Bordin, F

A. Bordin, F. J. B. Evertsz’, B. Roovers, J. D. T. Luna, W. D. Huisman, F. Zatelli, G. P. Mazur, S. L. D. ten Haaf, G. Badawy, E. P. A. M. Bakkers, et al., Probing Majorana Localization of a Phase-Controlled Three-Site Kitaev Chain with an Additional Quantum Dot, arXiv:2504.13702

arXiv
[28]

C.-X. Liu, A. M. Bozkurt, F. Zatelli, S. L. D. ten Haaf, T. Dvir, and M. Wimmer, Enhancing the Excitation Gap of a Quantum-Dot-Based Kitaev Chain, Commun. Phys. 7, 1 (2024)

2024
[29]

J. D. Torres Luna, A. M. Bozkurt, M. Wimmer, and C.-X. Liu, Flux-Tunable Kitaev Chain in a Quantum Dot Array, SciPost Phys. Core 7, 065 (2024)

2024
[30]

Miles, D

S. Miles, D. Van Driel, M. Wimmer, and C.-X. Liu, Kitaev Chain in an Alternating Quantum Dot-Andreev Bound State Array, Phys. Rev. B 110, 024520 (2024)

2024
[31]

Svensson and M

V . Svensson and M. Leijnse, Quantum Dot Based Kitaev Chains: Majorana Quality Measures and Scaling with Increasing Chain Length, Phys. Rev. B 110, 155436 (2024)

2024
[32]

Luethi, H

M. Luethi, H. F. Legg, D. Loss, and J. Klinovaja, Fate of Poor Man’s Majoranas in the Long Kitaev Chain Limit, Phys. Rev. B 111, 115419 (2025)

2025
[33]

R. A. Dourado, J. C. Egues, and P. H. Penteado, Two-Site Kitaev Sweet Spots Evolving into Topological Islands, arXiv:2501.19376

arXiv
[34]

C.-X. Liu, H. Pan, F. Setiawan, M. Wimmer, and J. D. Sau, Fusion Protocol for Majorana Modes in Coupled Quantum Dots, Phys. Rev. B 108, 085437 (2023)

2023
[35]

Pandey, S

B. Pandey, S. Okamoto, and E. Dagotto, Nontrivial Fusion of Majorana Zero Modes in Interacting Quantum-Dot Arrays, Phys. Rev. Res. 6, 033314 (2024)

2024
[36]

Tsintzis, R

A. Tsintzis, R. S. Souto, K. Flensberg, J. Danon, and M. Leijnse, Majorana Qubits and Non- Abelian Physics in Quantum Dot–Based Minimal Kitaev Chains, PRX Quantum 5, 010323 (2024)

2024
[37]

Boross and A

P. Boross and A. Pályi, Braiding-Based Quantum Control of a Majorana Qubit Built from Quantum Dots, Phys. Rev. B 109, 125410 (2024)

2024
[38]

D. M. Pino, R. S. Souto, and R. Aguado, Minimal Kitaev-Transmon Qubit Based on Double Quantum Dots, Phys. Rev. B 109, 075101 (2024)

2024
[39]

H. Pan, S. Das Sarma, and C.-X. Liu, Rabi and Ramsey Oscillations of a Majorana Qubit in a Quantum Dot-Superconductor Array, Phys. Rev. B 111, 075416 (2025)

2025
[40]

E. Zhuo, X. Yang, Y . Huang, Z. Lyu, A. Li, B. Li, Y . Zhang, X. Wang, D. Wang, Y . Shi, et al., Read out the Fermion Parity of a Potential Artificial Kitaev Chain Utilizing a Transmon Qubit, arXiv:2501.13367

arXiv
[41]

O. A. A woga and J. Cayao, Identifying Trivial and Majorana Zero-Energy Modes Using the Majorana Polarization, Phys. Rev. B 110, 165404 (2024)

2024
[42]

Bargerbos, M

A. Bargerbos, M. Pita-Vidal, R. Žitko, J. Ávila, L. J. Splitthoff, L. Grünhaupt, J. J. Wesdorp, C. K. Andersen, Y . Liu, L. P. Kouwenhoven, et al., Singlet-Doublet Transitions of a Quantum Dot Josephson Junction Detected in a Transmon Circuit, PRX Quantum 3, 030311 (2022)

2022
[43]

Bordin, F

A. Bordin, F. J. Bennebroek Evertsz’, G. O. Steffensen, T. Dvir, G. P . Mazur, D. Van Driel, N. Van Loo, J. C. Wolff, E. P. A. M. Bakkers, A. L. Yeyati, et al., Impact of Andreev Bound States within the Leads of a Quantum Dot Josephson Junction, Phys. Rev. X 15, 011046 (2025)

2025
[44]

Benestad, A

J. Benestad, A. Tsintzis, R. S. Souto, M. Leijnse, E. Van Nieuwenburg, and J. Danon, Machine- Learned Tuning of Artificial Kitaev Chains from Tunneling Spectroscopy Measurements, Phys. Rev. B 110, 075402 (2024)

2024
[45]

Luethi, H

M. Luethi, H. F. Legg, D. Loss, and J. Klinovaja, Properties and Prevalence of False Poor Man’s Majoranas in Two- and Three-Site Artificial Kitaev Chains, arXiv:2504.06732

arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A. Yu. Kitaev, Fault-Tolerant Quantum Computation by Anyons, Ann. Phys. 303, 2 (2003)

work page 2003

[2] [2]

Nayak, S

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80, 1083 (2008)

work page 2008

[3] [3]

S. D. Sarma, M. Freedman, and C. Nayak, Majorana Zero Modes and Topological Quantum Computation, Npj Quantum Inf. 1, 1 (2015)

work page 2015

[4] [4]

A. Y . Kitaev, Unpaired Majorana Fermions in Quantum Wires, Phys.-Uspekhi 44, 131 (2001)

work page 2001

[5] [5]

Fu and C

L. Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett. 100, 096407 (2008)

2008

[6] [6]

Leijnse and K

M. Leijnse and K. Flensberg, Introduction to Topological Superconductivity and Majorana Fermions, Semicond. Sci. Technol. 27, 124003 (2012)

work page 2012

[7] [7]

Sato and Y

M. Sato and Y . Ando, Topological Superconductors: A Review, Rep. Prog. Phys. 80, 076501 (2017)

work page 2017

[8] [8]

R. M. Lutchyn, E. P. a. M. Bakkers, L. P . Kouwenhoven, P. Krogstrup, C. M. Marcus, and Y . Oreg, Majorana Zero Modes in Superconductor–Semiconductor Heterostructures, Nat. Rev. Mater. 3, 52 (2018)

work page 2018