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Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
by Takuya Okuda, Aswin Parayil Mana, Hiroki Sukeno
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Submission summary
| Authors (as registered SciPost users): | Takuya Okuda · Aswin Parayil Mana · Hiroki Sukeno |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202406_00026v1 (pdf) |
| Date submitted: | 2024-06-12 16:48 |
| Submitted by: | Parayil Mana, Aswin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Calderbank-Shor-Steane (CSS) codes are a class of quantum error correction codes that contains the toric code and fracton models. A procedure called foliation defines a cluster state for a given CSS code.We use the CSS chain complex and its tensor product with other chain complexes to describe the topological structure in the foliated cluster state, and argue that it has a symmetry-protected topological order protected by generalized global symmetries supported on cycles in the foliated CSS chain complex. We demonstrate the so-called anomaly inflow between CSS codes and corresponding foliated cluster states by explicitly showing the equality of the gauge transformations of the bulk and boundary partition functions defined as functionals of defect world-volumes. We show that the bulk and boundary defects are related via measurement of the bulk system. Further, we provide a procedure to obtain statistical models associated with general CSS codes via the foliated cluster state, and derive a generalization of the Kramers-Wannier-Wegner duality for such statistical models with insertion of twist defects. We also study the measurement-assisted gauging method with cluster-state entanglers for CSS/fracton models based on recent proposals in the literature, and demonstrate a non-invertible fusion of duality operators. Using the cluster-state entanglers, we construct the so-called strange correlator for general CSS/fracton models. Finally, we introduce a new family of subsystem-symmetric quantum models each of which is self-dual under the generalized Kramers-Wannier-Wegner duality transformation, which becomes a non-invertible symmetry.
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 #2 by Anonymous (Referee 2) on 2024-9-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202406_00026v1, delivered 2024-09-07, doi: 10.21468/SciPost.Report.9720
Strengths
1. The paper successfully connects several important concepts in modern condensed matter theory, including fracton models, SPT order, anomaly inflow, and dualities.
2. The authors provide detailed mathematical formulations, particularly in their use of chain complexes and their tensor products to describe foliated cluster states. This formulation provides versatility in describing a variety of different topological phases.
3. The paper presents several new and important results, including: A generalized anomaly inflow mechanism for CSS codes; An extension of the Kramers-Wannier-Wegner duality for a broad class of models; A general construction of strange correlators for CSS codes; Identification of non-invertible symmetries in self-dual models.
Weaknesses
1. While the mathematical rigor is a strength, it may also make the paper challenging for readers not deeply familiar with algebraic topology and homological algebra. Some additional explanatory text or intuitive descriptions could help broaden the paper's audience.
2. The paper is quite lengthy. Therefore, sometimes it is hard to comprehend the connection between different parts of the paper. The authors have a brief discussion of summary of results in the beginning. But a expansion of this part can better guide the readers.
3. The definition of strange correlator does not involve the notion of duality. In Section 4, the authors seem to make a connection between these two ideas. Can the authors elaborate the motivation for such connections preferably in the beginning of the section?
Report
This paper represents a solid contribution to the field, providing a unified framework for understanding a wide range of phenomena related to CSS codes and fracton models. The work is mathematically rigorous and presents several novel and important results. The overall quality and importance of the work are high. However, the presentation could be improved to enhance accessibility. Therefore, I would recommend publication of this paper possibly after some revisions to improve the presentation.
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 1) on 2024-8-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202406_00026v1, delivered 2024-08-06, doi: 10.21468/SciPost.Report.9540
Report
This paper studies the foliated construction of CSS codes first proposed in Phys. Rev. Lett. 117, 070501 [1]. While [1] mostly focused on the error correction properties of the codes, this paper discusses various physical properties of the constructed codes, including their potential SPT order, anomaly inflow, duality, and BF theory description.
The paper pointed out some potentially interesting features of the constructed code. For example, while the system with periodic boundary has unique ground state (as a cluster state), with open boundary in the foliation direction, some bulk stabilizers reduce to logical operators of the boundary CSS code and may anti-commute with each other. In this sense, the cluster state has 1d SPT order under Z2xZ2 (or copies of it) symmetry. While this is a very simple type of SPT order, it may be of interest to the QI community. Similarly, the paper discussed the idea of anomaly inflow, duality, strange correlator and field theory description for these models. While it is not immediately clear what these features can be useful for, it may help the QI community better learn these physics ideas which are usually discussed in a very different language.
I think this paper is a good addition to the literature and probably above the bar for publication in SciPost Physics. However, I think a major weakness of the paper is that it started the discussion from the very beginning using a sophisticated chain-complex formalism. The chain-complex language is a powerful language for discussing CSS codes, but not every one is familiar with it. To make things worse, notations are not always well-defined. For example, section 2.2 started by talking about "objects assigned to qubits i". It is not clear at all what these ojects are referring to. Basic terminologies associated with chain complex, like Ker, Im, and Hom, are not defined or explained. I think the paper will be much more readable if the authors do a more careful job explaining these ideas and illustrate them immediately using a simple example. It would be very helpful if the authors can use one (or two) example to illustrate the various ideas introduced in the paper (SPT order, anomaly inflow, duality etc). Right now, examples are clustered in later sections and different parts use different examples.
Overall, I think the result is worth publishing, but if the authors want more people to actually read it, it needs to be written in a much better way.
Recommendation
Ask for major revision