Pareto-optimality of Majoranas in hybrid platforms

Luna et al. theoretically investigate the simultaneous optimization of localization length and topological gap. They perform this analysis for both hybrid nanowires and hybrid quantum dot chains. To achieve this, the authors utilize multi-objective optimization and, in the case of hybrid quantum dot chains, a perturbation theory that effectively allows for longer-range inter-dot coupling. The authors’ main claim is that hybrid quantum dot chains can realize an optimal regime characterized by short localization lengths and substantial gaps. They argue that an equivalent regime is not realized in nanowires because it would require very small Fermi velocities, thereby necessitating the system to be tuned close to the bottom of the semiconductor band.

I consider the research concept solid, and several of the results are interesting. For example, the observation that longer-range couplings arising from perturbation theory result in a finite localization length in the quantum dot chain. However, I cannot recommend publication in the current form. The conclusions appear to be largely the result of a comparison between realistic nanowire parameters and unrealistic quantum dot chain parameters, specifically regarding $\Delta$. Furthermore, there are numerous statements that lack clarity. I could see a modified version of the manuscript being appropriate for SciPost Physics, but in its current form, I fear the manuscript is very likely misleading to readers.

I outline my specific concerns and questions below:

1) My primary concern is that Figure 5 (which presents the main result) is misleading. Specifically, the parameter $\Delta$ carries a significantly different meaning for the two platforms. For the nanowire, it represents the induced gap due to the parent superconductor. However, for the quantum dot chain, it represents the pairing amplitude of the ABS. Even though $\Delta$ is, therefore, a physically distinct quantity (if I understand Table 1 correctly), the value of $\Delta$ in Figure 5 is chosen to be identical for both the nanowire and the chain. This renders the comparison somewhat meaningless; for instance, an ABS with a pairing potential equal to the induced gap (while also possessing a finite ABS energy and Zeeman energy) does not appear realistic.

The authors should comment on this distinction and, for a fair comparison, they should use an ABS pairing amplitude that is actually achievable in experiments, rather than the parent gap. It should also be discussed that $\Delta$ for the ABSs will likely not be constant as a function of other parameters, e.g., $\mu_A$.

2) The authors do not seem to make a clear distinction between a localized bound state and a Majorana bound state. For instance, in Figure 4, the authors state that the plot shows "Majorana quality in the topological phase," but in the current version, it is unclear how the authors determine if the system is topological, i.e., what invariant is being used?

I believe this is made more confusing by the following points: a) The limit $t=0$ is included in these plots as being in the "topological phase," yet in this limit, the chain consists merely of localized fermions. b) The caption refers only to $\delta\mu^{(2)}$, but the quantity defined in Eq. (16) is site-dependent (with an index $i$ that is missing in the caption). Does this imply that $\mu_D$ is now site-dependent? c) I do not have a clear intuition for the two-site sweet spot line in this figure. Why, in these units, does it not start at $\tilde \mu=0$? d) It is unclear what parameters were used in this figure 4, e.g., for $E_Z$, $\theta$, and $\Delta$. It would be useful to include these in the caption to make the magnitude of $E_{gap}$ meaningful.

3) The authors make several statements about disorder in QD chains that appear too general. For example: "the QD chain approach enables tuning away disorder." While I agree that this setup allows for the tuning of potential disorder, this will not be the case for disorder in other parameters, such as g-factors, $\Delta$, $\theta$, etc., all of which can vary for QDs.

4) In the discussion of "quality metrics," it appears the authors are missing a discussion regarding the size of the topological phase. If one wishes to perform manipulations, one requires a phase region that is sufficiently large; otherwise, one risks exiting the topological phase. Although related, I note that this is not identical to maximizing the gap or avoiding Landau-Zener transitions. For instance, in a two-site Kitaev chain at the sweet spot, the gap might be relatively large and the localization length ideal (a single site). However, if a slight deviation in parameters causes the system to quickly exit the topological phase, this would be detrimental for any potential qubit.

5) For Figure 7 (Appendix D) it is stated that magnetic B "saturates at the maximum allowed value" I did not quite understand what this meant. I note that $\Delta$ will likely depend on $B$ in a manner that is not captured by the simple renormalization in Appendix C.

Note: The caption in Figure 7 does not correspond to the figure itself. It states: "In panels (c-f) we show the remaining microscopic parameters," yet panels e-f do not exist.

I find the authors' research idea interesting, but the issues raised above mean that I cannot currently recommend publication. In particular, I feel that Point 1 renders the main comparison misleading. As such, additional caveats and clarifications must be added before I would feel comfortable recommending this manuscript for publication in the current form.

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