Symmetry-protected gates of Majorana qubits in a high-Tc superconductor platform

In their work, the authors propose a way to perform arbitrary quantum gates by manipulating Majorana corner states in a second-order topological superconductor. The starting point is a time-reversal invariant second-order topological superconductor as proposed in Refs. [61,62] of the manuscript. In such a system, each corner of a rectangular sample hosts a Kramers pair of Majorana corner states. When an additional in-plane field is applied, time-reversal symmetry is broken and the Kramers partners can split in space. Importantly, however, the presence of an additional emergent time-reversal symmetry still prevents the two Majoranas at a given corner from hybridizing. The authors show that changing the magnetic field along certain closed paths in parameter space gives rise to non-Abelian Berry phases that realize non-trivial rotations in the degenerate ground state manifold accompanying the Majorana corner states. It is proposed to use these rotations to perform quantum gates on qubits encoded within the degenerate ground state manifold.

The manuscript addresses a timely topic, is well-written, and appears to be technically sound. There are, however, several points that I hope the authors can comment on further. In particular, I have some concerns regarding the experimental feasibility and stability of the proposed manipulations. Furthermore, there are also some technical points that would benefit from an extended discussion.

Explicitly, the points I would like to raise are the following:

1. While the authors advertise their proposal as experimentally feasible and advantageous compared to several other computational schemes, there are a number of issues that might arise in an experimental realization of the proposed setup:

1.1. The authors give a detailed discussion of the symmetries required to protect the Berry phases under consideration. However, these symmetry constraints appear to be very stringent (see, e.g., the conditions (1)-(3) listed on pages 18 and 19 of the manuscript) and I am wondering how well they will be fulfilled in any experimental implementation of the proposed setup. In particular, it seems that an out-of-plane component of the Zeeman field would immediately break the emergent time-reversal symmetry required to protect the proposed phase gates. Given the extensive (and experimentally non-trivial) manipulations of the in-plane Zeeman fields required to achieve the proposed quantum gates, it seems extremely challenging to completely avoid the accidental presence of a small out-of-plane component of the Zeeman field. This is even more so, since the Zeeman fields have to be local and are assumed to be non-zero only at a given corner where the manipulation is supposed to take place. But how is this possible with only in-plane B-fields which also need to satisfy div B=0?

1.2. The authors assume perfect experimental control over the applied Zeeman fields. In particular, it is assumed that the path along which the Zeeman field evolves in the (B_x,B_y) plane is always a closed one. However, what if---by accident---one does not end up at the same point in parameter space from which one has started, i.e., the path in parameter space is not closed? If the experimental control of the Zeeman field is not perfect, such errors could easily go unnoticed. There could also, for example, be a danger of ‘overshooting’ the magnetic field and performing a 2\pi\pm\epsilon rotation instead of an exact 2\pi rotation (in case of a full braid). How would this affect the results presented in the manuscript, and are there ways to detect, remedy, or avoid such errors?

1.3. One should mention that a d-wave superconductor by itself already presents a highly non-trivial state of matter. It would be interesting to see if similar results could also be obtained from a time-reversal invariant topological insulator engineered from conventional ingredients only (see, e.g., Phys. Rev. B 102 (19), 195401).

2. When the model is studied in bosonic language, the considered situation is slightly different from the initial situation studied in the fermionic case, see Fig. 2. In the latter case, the length l depends on the strength of the Zeeman field and goes to zero as the Zeeman field goes to zero. In the model that is studied in the bosonic language, however, the length l is fixed and stays finite at all times. This is crucial for the expression for the level spacing given in Eq. (46) and as such also for stability of the proposed ‘half-moon’ and ‘slice-of-pie’ contours where the Zeeman field goes through zero at some point. The authors should comment on the differences between the two models studied in the fermionic and the bosonic language. Do these differences affect the results that are obtained and if so, in what way? Which situation is closer to the one actually encountered in an experimental realization of the model?

3. In the case of the ‘half-moon’ and ‘slice-of-pie’ contours, the Zeeman field goes to zero at some point of the process. In this case, the gap separating the ground state manifold from the excited states is given by the finite-size level spacing, see Eq. (46). I would expect that this gap can become quite small, which would then again put strong constraints on the adiabaticity of the process. Can the authors give an estimate on the energy scales involved in a realistic implementation of their scheme, and specify what this means for the adiabaticity condition of the process?

4. The intuition behind the ‘braiding’ process via a 2pi rotation of the in-plane Zeeman field could be made clearer. As far as I understand, there is no actual spatial exchange of the two Majoranas. Rather, the process seems to be closely connected to the one discussed in arXiv:1803.02173 (Ref. [58] in the manuscript). What is the intuition behind calling such a process an actual ‘braiding’ process? It would greatly benefit the accessibility of the manuscript if the authors could add some additional explanations here.

5. It should be emphasized more clearly that some of the proposed manipulations (in particular, the arbitrary phase gates via ‘slice-of-pie’ contours) do not correspond to quasiparticle braiding processes. While the introduction correctly states that it is fundamentally impossible to realize non-Clifford gates by braiding Majoranas, this distinction is somewhat blurred in the main part of the manuscript. This is also due to the fact that even the single exchange (‘half-moon’ contour) does not correspond to a standard exchange of Majorana quasiparticles as one would typically have in mind when talking about topological quantum computation by non-abelian anyons. As the magnetic field goes to zero during the exchange process, the two Majoranas strongly overlap and the inherent topological protection accompanying a set of well-separated Majoranas is lost. What one is left with is merely a symmetry-protected non-Abelian Berry phase that could emerge also in any other two-level system (not based on anyonic quasiparticles) in the presence of suitable symmetries. Overall, the processes discussed in the manuscript at hand seem to be closely related to holonomic quantum computation. The authors should comment on these connections more transparently.

6. The results on the parafermionic case come with some unresolved problems such as unwanted dynamical phases. While this is correctly pointed out in the main text, I recommend that the corresponding part of the abstract should be phrased more carefully. In particular, the sentence “Our analysis naturally includes interaction effects and can be generalized to cases with fractional bulk excitations” tends to oversell the results on the parafermionic case.

7. Finally, I have a few more detailed remarks regarding specific parts of the manuscript:

7.1. Fig. 1: I do not understand the meaning of the red oval in the drawing. Does it represent a path in parameter space (=direction of magnetic field)?
7.2. Page 6, in between Eqs. (2) and (3): I am not sure I understand what is meant by the remark ‘We choose x to be along the diagonal direction.’ From the equations, I would assume x to be an edge coordinate running along the edge of a finite sample. It would be helpful to indicate the coordinate axes in Fig. 1, for example.
7.3. In Eq. (9): x_0 is not defined. Should it be x_-?
7.4. On page 7, after Eq. (9), it reads: “[…] there exist a pair of MZMs separated by a length l, the length of the region where |\Delta|=B.” Should it be |\Delta|<B?
7.5. Eq. (15): Is the complex conjugation on the RHS correct? It is already included in the definition of P.
7.6. Page 8, after Eq. (16): I believe that \bar{C} is not defined.
7.7. Eq. (18b): It should be x’ instead of x in the second argument on the LHS.
7.8. Eqs. (55), (56), and (57): I do not understand the difference between the \psi in the bra and \Psi in the ket. Is it a different state?
7.9. Page 18, condition (2): I believe there is a typo – the same condition Ba>>(a/l)^{2-K} is given twice. The same question also applies to page 19, again condition (2).
7.10. Eq. (68): The use of \tau_0 is somewhat unfortunate here, since the same symbol is also used for the zeroth Pauli matrix in particle-hole space [see, e.g., Eq. (10)].
7.11. Eq. (70): I do not understand the meaning of (and the difference between) k_a, k_b, k_x, and k_y.
7.12. Page 23, paragraph after Eq. (71): H_B is not defined. Should it be H_Z?

The authors study an effective edge theory of a heterostructure consisting of a QSH insulator and a d-wave superconductor. The analyze the effective symmetries of the theory and the existence of corner Majorana zero modes in the presence of domain walls caused by local Zeeman fields. Then, they propose a way to implement symmetry-protected quantum gates via the rotations of the Zeeman field and its effect on the position of the MZM. Interestingly they point out that non-Clifford rotations can be implemented via a "slice of pie" contour in the (Bx, By) plane. The formalism used allow generalization to the fractional case, and several lengthy appendices quench the thirst for technical details.

This work is interesting and timely given recent interest on heterostructures of the type studied here. Indeed, the authors point out a few explicit material examples (WTe and BSCCO), although their theoretical analysis is quite far from realistic samples. On the technical side, the zero-mode analysis is carried out with lucidity and using known techniques, which leaves little doubt as to the correctness of their results. In fact, the effective edge theory is that of a QSH edge subject to local Zeeman and s-wave pairing (with possible sign changes in Delta inherited by the parent d-wave pairing), which is familiar territory.

I find that this work meets one of the expectations for acceptance in SciPost (namely, "Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work") and has the clear potential to meet all of the required general acceptance criteria. Thus, I believe that this work should be eventually be published in SciPost Physics. I have some remarks on the clarity of the manuscript which I think deserve further edits and review before publication.

First, I find that the setup and coordinate system should be better described.

• How exactly does the setup in Fig. 1 avoid the problem of single-particle tunneling from nodal gapless states in the parent superconductor?
• What is the coordinate system used? Is x a coordinate that parametrizes the position along the edge in Fig.1 (including the corner), or is the (x,y) coordinate system fixed with respect to the sample? It is not always clear to follow the conventions adopted throughout the text, especially when it comes to the field labeling.

Second, the simplicity and robustness of the proposal are somewhat oversold and should be discussed more critically.

• Most importantly, I do not understand the robustness of the "pie slice" protocol in Fig. 4. Even if all required symmetry conditions are satisfied, it seems to me that a small error in implementing the angle tau0 will result in a linear error in the final phase gate of Eq. 60. So how can the Berry phase be robust to small deformations of the arc? Practically speaking, what is the advantage with respect to a poor-man's rotation implemented via the timing of an interaction between MZMs?
• The effective time-reversal symmetry T_tilde seems to be crucial for protected operations. What are examples of perturbations that break this symmetry? Can they be expected in "real" devices?
• The entire scheme seems rely on the absence of disorder and on the presence of several crystalline symmetry. Can it survive against, say, irregularities or non-uniformity in the sample shape? For instance, it seems to me that a certain relative orientation between the QSH insulator and the d-wave superconductor must be required to avoid the interaction between the nodal gapless excitations, but this aspect, while mentioned, is not clearly illustrated and looks worryingly fragile.

Finally, the clarity of the presentation would benefit from one thorough proof-reading. Overall, the style and length of the work are appropriate and pleasant, but the notation is inconsistent or hard to trace at times, an