Machine-learned tuning to protected states by probing noise resilience

Technical Summary: Machine-learned tuning to protected states by probing noise resilience

Problem Statement
Protected quantum states, such as Majorana bound states (MBSs) in topological superconductors, are essential for fault-tolerant quantum computation due to their intrinsic resilience against specific noise channels. However, these states typically exist only at discrete points or small regions within a high-dimensional parameter space (often referred to as "sweet spots"). Locating these regimes experimentally is challenging because standard tuning metrics (e.g., local conductance measurements) only indirectly reflect the formation of topologically non-trivial states and provide limited evidence of their presence. Existing tuning protocols often rely on complex, indirect strategies that struggle to navigate the parameter space efficiently.

Methodology
The authors propose a direct machine-learning approach to tune systems to protected regimes by utilizing the system's intrinsic noise resilience as the primary optimization metric. The method employs the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) to minimize the sensitivity of the ground-state energy splitting ($E_0$) to random parameter fluctuations.

The specific implementation focuses on quantum dot (QD)-based Kitaev chains:

1. System Model: The study considers a chain of $N = 2n_s - 1$ QDs, where even sites are proximitized by superconductors and odd sites are normal. The Hamiltonian includes onsite potentials ($\varepsilon_i$), Zeeman fields ($V_{z,i}$), induced superconducting pairing ($\Delta_i$), spin-conserving hopping ($t_i$), and spin-orbit coupling ($t^{so}_i$).
2. Optimization Loop:
   • The algorithm generates a population of tuning configurations (sets of onsite potentials $\varepsilon$).
   • For each configuration, the system is subjected to random local noise (fluctuations $\eta$) drawn from a uniform distribution $[-W, W]$ applied to all QD levels.
   • A loss function $L$ is calculated as the average ground-state energy splitting $E_0$ resulting from these fluctuations.
   • The CMA-ES algorithm updates the distribution of parameters to minimize this loss, effectively searching for the configuration where $E_0$ is least sensitive to noise.
3. Verification: The robustness of the method is tested across varying chain lengths (2 to 5 sites), the inclusion of electron-electron interactions, asymmetric coupling setups, and finite Zeeman fields.

Key Results

• 2-Site Chains: In the shortest system capable of supporting localized MBSs ($N=3$ QDs), the algorithm consistently converged to specific "sweet spots" in parameter space across 50 independent runs. At these points, the ground-state splitting $E_0$ was minimized, the Majorana polarization (MP) at the outer dots approached unity ($|M| \approx 1$), and a finite excitation gap ($E_{ex}$) was maintained.
• 3-Site Chains: For longer chains ($N=5$), the method successfully identified sweet spots. The authors demonstrated control over the trade-off between MP and the excitation gap by adjusting the noise amplitude on the outermost dots relative to the center. Increasing the noise penalty on outer dots ($\beta > 1$) led to configurations with larger excitation gaps but slightly reduced MP.
• Longer Chains: The protocol was extended to 4-site ($N=7$) and 5-site ($N=9$) chains. While the method successfully minimized $E_0$ and converged to protected states, the excitation gap showed more variability compared to shorter chains, suggesting that fine-tuning phases in longer arrays is more complex.
• Robustness: The results remained qualitatively unchanged when electron-electron interactions were included or when the system was modeled with asymmetric couplings, reflecting realistic experimental conditions.

Significance and Claims
The paper claims to provide a reliable, general method for tuning to protected states by directly probing their resilience against noise, rather than relying on indirect signatures.

• Directness: The approach bypasses the need for complex conductance measurements or indirect topological invariants, using only the ground-state energy splitting ($E_0$) as the feedback signal.
• Generality: While illustrated on Kitaev chains, the authors assert the method is applicable to a broad class of platforms supporting different types of protected states, whether topological or non-topological.
• Experimental Feasibility: The method is presented as readily applicable to experiments, as the required measurement (ground-state energy splitting) can be obtained via conductance spectroscopy or Ramsey-type experiments in qubit setups.
• Scope: The work emphasizes that the method is not limited to isolated MBSs but serves as a general strategy for finding any protected regime where noise resilience is a defining characteristic.