Braiding Majoranas in a linear quantum dot-superconductor array: Mitigating the errors from Coulomb repulsion and residual tunneling

1. Problem Statement

The paper addresses the challenge of experimentally demonstrating non-Abelian exchange statistics of Majorana zero modes (MZMs) in a realistic, engineered platform. While topological superconductors (specifically Kitaev chains) are theoretically robust, practical implementations using quantum dot-superconductor arrays face significant physical imperfections that degrade the fidelity of braiding operations.

The specific challenges identified are:

• Interdot Coulomb Repulsion: Unlike idealized models, quantum dots exhibit strong long-range Coulomb interactions between the ancillary dot and the chain dots. This shifts energy levels and prevents the system from reaching the necessary resonance conditions for braiding.
• Residual Single-Electron Tunneling: In real devices, electrostatic gates cannot perfectly turn off tunneling couplings ($\Gamma$). These residual couplings introduce "leakage" errors (populating excited states) and "geometrical" errors (deviations in the Berry phase).
• Phase Sensitivity: Successful braiding requires a precise superconducting phase difference of $\phi = \pi$. Deviations or noise in this parameter can destroy the topological protection.

The authors aim to determine if a minimal braiding protocol (using a linear array rather than a complex T-junction) can be made robust against these specific quantum dot imperfections.

2. Methodology

The authors propose and simulate a minimal braiding setup consisting of:

• Architecture: Two copies of a two-site Kitaev chain (Left and Right) connected via a central ancillary quantum dot (Dot D).
• Hamiltonian: A full microscopic model including:
  • Kitaev chain terms (chemical potential $\mu$, hopping $t$, superconducting pairing $\Delta$).
  • Tunneling between the chain ends and the ancillary dot ($\Gamma_L, \Gamma_R$).
  • Interdot Coulomb interaction ($U_L, U_R$).
  • Phase difference $\phi$ controlled by magnetic flux.
• Protocol: A double-braid sequence (6 steps) involving the cyclic tuning of the ancillary dot's chemical potential ($\mu_D$) and the tunnel couplings ($\Gamma_L, \Gamma_R$). This sequence effectively creates a "virtual trijunction" where the ancillary dot mediates the exchange of two MZMs.
• Simulation & Analysis:
  • Numerical: Time-dependent Schrödinger equation solving using QuTiP to calculate state evolution and infidelity ($1-F$).
  • Analytical: Perturbation theory and effective Hamiltonian derivation (Majorana basis) to understand error mechanisms (leakage vs. geometrical errors).
  • Noise Modeling: Quasi-static disorder approximation to simulate charge noise ($\delta\mu_D$) and tunneling noise ($\delta\Gamma$).

3. Key Contributions

A. Mitigation of Coulomb Repulsion

The paper demonstrates that strong interdot Coulomb repulsion ($U$) shifts the energy of the ancillary dot, preventing the system from reaching the resonance required for MZM exchange.

• Solution: The authors derive an analytical correction for the ancillary dot's chemical potential:
  $$\mu_{D, \text{min}} = \mu^*_D = \sum_{a=L,R} \left( -\frac{U_a}{2} - \Delta_a + \sqrt{\Delta_a^2 + (U_a/2)^2} \right)$$
• Result: By shifting the operating range of $\mu_D$ to this optimal value ($\mu^*_D$), the infidelity drops from $\approx 1$ (complete failure) to $< 10^{-3}$, even for strong Coulomb interactions ($U > \Delta$).

B. Mitigation of Residual Tunneling

Residual tunneling ($\Gamma_{\text{min}} > 0$) causes two types of errors:

1. Leakage Error: Caused by the system leaking into excited states due to finite energy gaps. This is mitigated by increasing the variation range of the ancillary dot's chemical potential ($\Delta\mu_D \gg \Gamma_0$).
2. Geometrical (Berry Phase) Error: Caused by the residual couplings altering the path in parameter space.

• Solution: The authors propose a counter-intuitive operating point where the minimum chemical potential of the ancillary dot is negative:
  $$\mu_{D, \text{min}} = -\sqrt{2}\Gamma_{\text{min}}$$
• Result: This specific tuning cancels the leading-order geometrical error, reducing infidelity quadratically with respect to the residual coupling strength.

C. Experimental Signatures and Verification

The paper proposes concrete experimental methods to identify the optimal operating regime and verify successful braiding:

• Local Conductance Spectroscopy: Measuring the conductance through the ancillary dot reveals a subgap peak minimum at $\mu_D = \mu^*_D$, serving as a "sweet spot" indicator.
• Infidelity Mapping: Scanning $\mu_{D, \text{min}}$ while performing the braiding protocol shows a sharp dip in infidelity at the optimal point.
• Phase Verification: The authors propose a 3-step experiment to distinguish the $4\pi$-periodic Josephson effect (signature of MZMs) from standard $2\pi$ oscillations, confirming the non-Abelian nature of the exchange.

4. Key Results

• Robustness: The proposed protocol remains highly effective ($1-F < 10^{-3}$) in the presence of strong Coulomb repulsion and residual tunneling, provided the ancillary dot is tuned to the derived optimal parameters.
• Noise Sensitivity:
  • Charge Noise ($\mu_D$): Highly detrimental. The disorder strength must be much smaller than the tunneling scale ($\sigma_{\mu_D} \ll \Gamma_0$) to resolve the infidelity dip.
  • Tunneling Noise ($\Gamma$): Less detrimental. Fluctuations in the "off" state are suppressed, and fluctuations in the "on" state primarily affect the energy gap rather than the braiding outcome.
• Adiabaticity: The protocol requires a time scale $T \sim 300 \hbar/\Gamma_0$ (approx. 20 ns for typical parameters) to satisfy adiabaticity while suppressing diabatic errors.
• Double Braid Signature: The most convincing experimental signature is the double braid result: starting in state $|ee\rangle$, a single braid creates a superposition, but a double braid (6 steps) deterministically transforms the system to the orthogonal state $|oo\rangle$. This outcome is robust against Rabi oscillations caused by phase noise.

5. Significance

• Feasibility of Linear Arrays: This work resolves a critical open question: whether braiding protocols designed for nanowires (T-junctions) are valid in quantum dot arrays with strong interactions. The answer is yes, provided specific control strategies are employed.
• Minimalist Approach: It demonstrates that a "minimal" setup (linear array + one ancillary dot) is sufficient for non-Abelian statistics, avoiding the complexity of fabricating physical T-junctions.
• Error Correction Strategy: The paper provides a blueprint for active error mitigation in topological quantum computing. Instead of requiring perfect hardware, it shows that software/parameter tuning (specifically of the ancillary dot) can compensate for intrinsic device imperfections like Coulomb repulsion and gate leakage.
• Experimental Roadmap: By proposing specific spectroscopy and infidelity measurements, the paper bridges the gap between theoretical proposals and current experimental capabilities in quantum dot-superconductor hybrid devices (e.g., InAs/Al systems).

In conclusion, the paper establishes that optimal control of an ancillary quantum dot can effectively neutralize the two most significant sources of error in quantum-dot-based Majorana braiding, paving the way for the first experimental demonstration of non-Abelian statistics in a linear array.