[Paper Review] Fusion protocol for Majorana modes in coupled quantum dots

Chun-Xiao Liu, Haining Pan+3 more|arXiv (Cornell University)|Dec 3, 2022

Topological Materials and Phenomena56 references4 citations

TL;DR

This paper proposes a robust fusion protocol to detect non-Abelian statistics in Majorana zero modes realized in a four-quantum-dot chain, using electrostatic gate control and quantum capacitance-based parity readout. The protocol avoids the need for floating superconducting islands, demonstrating resilience to diabatic, dephasing, and calibration errors—particularly showing that coupling calibration errors are more detrimental than on-site energy errors.

ABSTRACT

In a recent breakthrough experiment [Nature (London) 614, 445 (2023)], signatures of Majorana zero modes have been observed in tunnel spectroscopy for a minimal Kitaev chain constructed from coupled quantum dots. However, as Ising anyons, Majoranas' most fundamental property of non-Abelian statistics is yet to be detected. Moreover, the minimal Kitaev chain is qualitatively different from topological superconductors in that it supports Majoranas only at a sweet spot. Therefore, it is not obvious whether non-Abelian characteristics such as braiding and fusion can be demonstrated in this platform with a reasonable level of robustness. In this work, we theoretically propose a protocol for detecting the Majorana fusion rules in an artificial Kitaev chain consisting of four quantum dots. In contrast with the previous proposals for semiconductor-superconductor hybrid nanowire platforms, here we do not rely on mesoscopic superconducting islands, which are difficult to implement in quantum dot chains. To show the robustness of the fusion protocol, we discuss the effects of three types of realistic imperfections on the fusion outcomes, e.g., diabatic errors, dephasing errors, and calibration errors. We also propose a fermion parity readout scheme using quantum capacitance. Our work will shed light on future experiments on detecting the non-Abelian properties of Majorana modes in a quantum dot chain.

Motivation & Objective

• To demonstrate the non-Abelian fusion rules of Majorana zero modes in a minimal four-quantum-dot Kitaev chain platform.
• To overcome the challenge of detecting non-Abelian statistics in quantum dot chains, where floating superconducting islands (required in nanowire platforms) are experimentally infeasible.
• To develop a gate-based manipulation and quantum capacitance-based parity readout scheme that avoids mesoscopic superconducting islands.
• To evaluate the robustness of the fusion protocol under realistic imperfections: diabatic errors, dephasing, and calibration errors in couplings and on-site energies.

Proposed method

• Uses electrostatic gates to tune dot energies and couplings (normal and superconducting) between adjacent quantum dots to implement the fusion protocol.
• Employs a two-step fusion process: first, all dot energies are tuned to zero; second, couplings between the central dots (D2 and D3) are turned off to probe the resulting Fermion parity.
• Applies quantum capacitance measurements to read out the parity of the fused state, with the second derivative of energy w.r.t. gate voltage used to extract capacitance.
• Models the system using a four-dot Kitaev chain Hamiltonian with on-site energies, normal tunneling, and superconducting pairing terms.
• Analyzes the effects of errors via numerical simulations, including diabatic evolution, dephasing, and calibration errors in coupling and on-site energy parameters.
• Uses occupation number basis to diagonalize the Hamiltonian in even- and odd-parity subspaces to compute ground-state energies and quantum capacitance.

Experimental results

Research questions

• RQ1 Can the non-Abelian fusion rules of Majorana zero modes be detected in a quantum dot chain without floating superconducting islands?
• RQ2 How robust is the proposed fusion protocol to diabatic errors, dephasing, and calibration errors in experimental parameters?
• RQ3 What is the relative impact of calibration errors in on-site energies versus tunneling/superconducting coupling strengths on fusion fidelity?
• RQ4 Can quantum capacitance measurements reliably read out the Fermion parity of the fused state in the presence of realistic experimental imperfections?
• RQ5 Is the fusion outcome probabilistic or deterministic under the proposed protocol, and how does it compare to reference schemes?

Key findings

• The fusion protocol successfully detects the non-Abelian fusion rule σ×σ=I+Ψ via a probabilistic parity readout with an average of zero, distinguishing it from the deterministic reference protocol.
• The protocol is robust against diabatic and dephasing errors, with fidelity remaining above 90% under moderate error amplitudes.
• Calibration errors in couplings (t and Δ) have a significantly stronger adverse effect on fusion outcomes than errors in on-site energies (ε), with fidelity dropping below 80% at 0.4Δ₀ for coupling errors.
• Quantum capacitance measurements are highly robust to calibration errors in on-site energies (ε), but sensitive to errors in tunneling and pairing couplings (t and Δ), consistent with fusion protocol results.
• The initial state calibration errors in couplings are more detrimental than final state errors, highlighting the importance of precise control during the fusion process.
• Numerical results for quantum capacitance agree with analytical predictions in the error-free case, validating the theoretical framework.