SciPost Submission Page
Sequential Adiabatic Generation of Chiral Topological States
by Xie Chen, Michael Hermele, David T. Stephen
Submission summary
| Authors (as registered SciPost users): | Xie Chen · David Stephen |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2402.03433v1 (pdf) |
| Date submitted: | 2024-08-16 19:53 |
| Submitted by: | Stephen, David |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In previous work, it was shown that non-trivial gapped states can be generated from a product state using a sequential quantum circuit. Explicit circuit constructions were given for a variety of gapped states at exactly solvable fixed points. In this paper, we show that a similar generation procedure can be established for chiral topological states as well, despite the fact that they lack an exactly solvable form. Instead of sequentially applying local unitary gates, we sequentially evolve the Hamiltonian by changing local terms in one subregion and then the next. The Hamiltonian remains gapped throughout the process, giving rise to an adiabatic evolution mapping the ground state from a product state to a chiral topological state. We demonstrate such a sequential adiabatic generation process for free fermion chiral states like the Chern Insulator and the $p+ip$ superconductor. Moreover, we show that coupling a quantum state to a discrete gauge group can be achieved through a sequential quantum circuit, thereby generating interacting chiral topological states from the free fermion ones.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
This manuscript generalizes the previous unitary sequential circuits to the case of chiral phases of matter. The focus is on free-fermion states, which is motivated based on field-theoretic approaches and confirmed via lattice simulations. The authors also comment on how this can apply to interacting chiral states via a gauging procedure. Finally the authors point out subtleties related to exponential tails and truncations.
This is an interesting work and while the construction is natural and convincing, I would not have expected that one can prepare chiral states in this way (before seeing the solution). The idea of growing a system whilst preserving periodic boundary conditions is rather nice. It shows how chiral states fit into the idea of sequential unitary circuits (or Hamiltonian generalizations thereof), and it also opens some interesting follow-up directions related to the truncated versions of such circuits. This work thus manages to make some surprising connections to recent concepts of interest, and motivates potential follow-up works. In principle I would like to recommend publication, but there is one part of the paper where I am not yet fully convinced:
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.
In addition, I had a few smaller points for the authors' consideration:
* Fig 5 has no labels
* 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?
* 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 ?
* Sec 4 feels rather brief / ends abruptly. Worth adding an explicit example?
* Sec 6: typo "Fibbonnaci state"
* Sec 6: section ends very abruptly (especially with "etc")
* 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)
Recommendation
Ask for minor revision
Strengths
1- The manuscript contains a plausible strategy to prepare chiral topological states, a family that previous works could not address.
2- The claim is supported by a numerical simulation of a free fermion model, which convincingly establishes that there exist some models for which this strategy succeeds.
3- The manuscript is clearly written.
4- The connection to sequential circuits is discussed in Sec 5 and offers a number of interesting open problems.
Weaknesses
1- Since there exist efficient algorithms to compile time evolution into quantum circuits, it is not clear why adiabatic Hamiltonian evolution is fundamentally different from quantum circuits.
2- No comparison is attempted with other existing approaches to prepare free-fermion states.
3- A gap is demonstrated numerically for two different system sizes and it's claimed that it remains constant, but it is not explained why this should be the case.
4- Beyond the Hamiltonian path for the topological insulator, the discussion is rather high level and sketched.
Report
The manuscript is insightful and a creative point of view on an important problem. As such it fulfils the SciPost acceptance criteria, especially #2: Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work.
I recommend acceptance, provided some of the manuscript shortcomings are addressed.
Requested changes
1- 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.
2- 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.
3- 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.
Recommendation
Publish (meets expectations and criteria for this Journal)