Sequential Adiabatic Generation of Chiral Topological States

We thank the referee for carefully reading our paper and for the helpful suggestions.

--- In Sec 4, the authors discuss gauging as a sequential circuit. This seems correct for the case that is written out, namely a bosonic Z_2 symmetry. But the authors want to use it for the case where Z_2 is fermion parity. There might be additional subtleties in this case, similarly to how the 2+1d fermionization map is subtle. Since this part of the paper is essential to making the case that one can build e.g. chiral Ising anyon theories, I can only recommend publication if the authors include the circuit for the case where the symmetry is fermion parity. Note that this requires some changes, since the controlled-Not gates would naively become fermionic gates which sounds problematic and requires additional care.

The generalization of the circuit in Fig. 9 to the fermionic case is straightforward. One just need to replace the $X$ in the controlled-Not operator (controlled by a $\tau_x$) with the fermionic parity operator $(-1)^{n_f}$. This controlled-$(-1)^{n_f}$ gate is a well-defined local, fermionic even, unitary gate acting in the gauge-matter Hilbert space. No issues with fermionization arise since we are dealing with physical fermionic degrees of freedom. Therefore, the replacement does not lead to any issue of non-commutativity among operators. We have added a paragraph at the end of section 4 to explain this point.


--- Fig 5 has no labels

For all the subfigures in Fig. 5, the y axis represents energy while the x axis represents momentum. To avoid redundancy in the plot, we put labels only in the first subfigure. We have added an explanation about this in the caption of Fig. 5.

--- Sec 4 (gauging as circuit): is it interesting to also add the extra FDLU step which disentangles "X prod tau^z = 1" --> "X = 1", so one is left over with just the gauge field?

One can add such a step. It simplifies the model but the topological order of the model remains the same. Therefore, to address the question we want to address in this paper, we do not think it is necessary to add this step.

--- Sec 4: "This circuit is very similar in form to the one discussed in [1] to generate the toric code state from the product state" -> should perhaps also cite https://arxiv.org/abs/2104.01180 ?

We thank the referee for the suggestion. The reference has been added.

--- Sec 4 feels rather brief / ends abruptly. Worth adding an explicit example?

We thank the referee for the suggestion. We added a paragraph now about generalizing the circuit to fermionic matter degrees of freedom. We also added some discussion of examples near the end of section 4.

--- Sec 6: typo "Fibbonnaci state"

Thanks, corrected.

--- Sec 6: section ends very abruptly (especially with "etc")

We have added a sentence "Therefore, this kind of circuit should be useful for further investigations of the gauging of many-body systems, including its implementation on quantum devices."

--- The abstract mentions "despite the fact that they lack an exactly solvable form" but that is not quite right since of course the main examples are exactly solvable (free-fermion)

Thanks for pointing this out. We made it more clear that we meant "a zero-correlation-length" exactly solvable form.

We thank the referee for carefully reading our paper and for the helpful suggestions.

--- The overall goal of the paper seems to be to provide an efficient way of preparing chiral topological states. In my opinion, the Introduction should briefly mention other approaches one could take, and discuss their shortcomings. If there are no competing approaches to prepare chiral states, this could also be mentioned.

The goal of this paper is not quite to provide an "efficient" way of preparing the chiral states. We want to answer the conceptual question of whether a sequential procedure can be used to generate the chiral state that's conceptually similar to the sequential circuit we used to generate non-chiral states. Special emphasis is put on ensuring the unitarity and locality of the protocol. If one were to focus on the efficiency of the protocol, one might consider using non-local gates or non-unitary transformations.

The only other protocol we are aware of is arxiv 2304.13748, which uses an RG like scheme to generate chiral states. As the length scale gets larger and larger in the RG protocol, the evolution involves nonlocal coupling terms. Because of this, we don't consider this other protocol as a competing protocol. But we did cite this paper in the introduction. We added a comment to make it clear that because of the involvement of the adiabatic evolution of nonlocal coupling terms, this protocol has a different setup than the one we are considering.


--- In the discussion on relation to quantum circuits, long-range/non-local interactions are mentioned. I think a bit of clarity could be added here by explaining that they arise merely due to moving from a torus to a flat 2D lattice.

We can think of three different potential generalizations to the torus (the cross section of which is illustrated in the attached figure), but none of them solves the nonlocality issues, each for different reasons.

In (a), the torus grows as the state is sequentially generated. The torus has to be embedded in 3d space with a uniform density of qubits per volume as a resource. The circuit has to act on the whole 2d torus at each sequential step (to move it outward at each step), moreover, the circuit winds up living in 3d instead of in 2d. So this becomes rather far from what we might mean by a 2d sequential circuit or sequential adiabatic evolution.

If we are only allowed to grow the torus at one location while keeping other parts intact, the geometry becomes (b). This procedure can be carried out strictly in 2d (in a bilayer 2d system), but one essentially gets a “folded” chiral state, because the top and bottom halves of the torus are close together. Therefore, we should think of this state as a stack of two states with opposite chiralities, and not a chiral state.

Finally, in (c), the state partially covers a large torus when it grows (indicated by the red arrow). We encounter the same nonlocality issue as on the 2d plane when the size of the state becomes a finite fraction of the torus.
We have added a comment at the beginning of section 5 to make it clear that we are considering a flat 2D plane geometry.


--- I don't understand the discussion around exponential tails potentially being an obstruction to preparation through circuits. Already from the MPS example it's clear that a sequential circuit can prepare states with any sort of exponential decay. Of course it's probably not true that a sequential circuit can produce any sort of correlations as long as they are exponentially decaying, but this is also not clear in sequential adiabatic preparation.

It is true that MPS with a finite correlation length can be generated with sequential circuits, so the finite correlation length of chiral states is not by itself a reason why a sequential circuit generation is not possible. We suspect that a sequential circuit generation would be impossible for chiral states because otherwise there would be a finite bond dimension tensor network representation of chiral states, which is generally believed not to be true. In any case, all we are saying here is that if we naively convert the adiabatic process into a circuit, it does not give a circuit with strictly local gates. We added a comment in the last paragraph that "exponentially decaying correlation alone does not forbid the possibility of sequential circuit generation. Indeed, it is known that matrix product states can all be generated using sequential circuits, even if they have nonzero correlation length."

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