SciPost Submission Page
Integrability of open boundary driven quantum circuits
by Chiara Paletta, Tomaž Prosen
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Chiara Paletta |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2406.12695v2 (pdf) |
| Date submitted: | 2024-07-19 18:17 |
| Submitted by: | Paletta, Chiara |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper, we address the problem of Yang-Baxter integrability of doubled quantum circuit of qubits (spins 1/2) with open boundary conditions where the two circuit replicas are only coupled at the left or right boundary. We investigate the cases where the bulk is given by elementary six vertex unitary gates of either the free fermionic XX type or interacting XXZ type. By using the Sklyanin's construction of reflection algebra, we obtain the most general solutions of the boundary Yang-Baxter equation for such a setup. We use this solution to build, from the transfer matrix formalism, integrable circuits with two step discrete time Floquet (aka brickwork) dynamics. We prove that, only if the bulk is a free-model, the boundary matrices are in general non-factorizable, and for particular choice of free parameters yield non-trivial unitary dynamics with boundary interaction between the two chains. Then, we consider the limit of continuous time evolution and we give the interpretation of a restricted set of the boundary terms in the Lindbladian setting. Specifically, for a particular choice of free parameters, the solutions correspond to an open quantum system dynamics with the source terms representing injecting or removing particles from the boundary of the spin chain.
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
In this paper, the authors investigated a special type of integrable quantum circuits with open boundaries. The bulk fundamental gate is the tensor product of a pair of two qubit gates which satisfy the Yang-Baxter equation. The gates at the boundary is required to satisfy Sklyanin's reflection algebra, or boundary Yang-Baxter equation so that the whole system is integrable. They ask the question whether the boundary K-matrix can have a non-factorized solution. By solving the reflection algebra explicitly, they show that non-factorized solutions only exist when the bulk gate corresponds to the $R$-matrix of the free XX spin chain. In this case, and for specific choices of boundary parameters, the circuit system can be viewed as a discretized Lindbladian evolution system.
This work is interesting enough to be published on SciPost. The paper is clearly written and reviewed proper amount of contents. I only have some minor comments.
1. I think some brief review of integrable Lindblad evolution system could be helpful for the reader. The reason is that without this context or background, it is slightly difficult to appreciate why the authors consider the double copy system and investigate the specific question of factorizability of the boundary K-matrix.
2. The conclusion of this paper, namely only the free bulk theory can give rise to the non-factorized boundary K-matrix reminds me some very similar results in the context of integrable quantum field theory (IQFT). For an IQFT, one can consider defects and there is a notion of an integrable defect, defined Delfino, Mussardo and Semonetti (see hep-th/9409076). There are two types of integrable defects, which are the topological and non-topological defects. The topological defects are purely transmissive while the non-topological ones are simultaneously transmissive and reflective. In the work of Catro-Alvaredo, Fring and Gohmann (see hep-th/0201142), they showed that the only theories that allows non-topological integrable defects are the free theories (free fermion or free boson). It is known that the defects can be related to the boundary by the so-called “folding trick”. Roughly speaking, “folding” the theory along the defect gives a boundary. It seems to me that the folding trick gives two copies of the bulk theory naturally and the distinction between topological and non-topological defects are related to the factorizability of the boundary S-matrices. The two conclusions are very similar and might actually be related in some cases. It would be interesting if the authors could make some comments on this.
Requested changes
1. Give a brief review on integrable Lindblad evolution system;
2. Comment on the point which I raised in the report (this is not compulsory).
Recommendation
Ask for minor revision
Strengths
1. Timely and interesting topic
2. Developing a general framework
3. Clear and well-written exposition
Weaknesses
None
Report
The paper develops a framework for constructing Yang-Baxter integrable open systems with Lindblad jump operators located at the boundary. This framework is especially powerful given its generality, and in my view, the most exciting conclusion is that all such systems correspond to free fermionic dynamics in the bulk. I have no hesitation in recommending this work for publication in Scipost Physics.
Requested changes
None
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1-Interesting new results by classifying quantum circuits with open boundaries
2-Clearly written
Weaknesses
1-Not too much of the physical properties of the new models are investigated and it's not clear if they give some interesting physics.
Report
This paper looks for integrable quantum circuits with open boundary conditions. The authors consider a set-up where they consider the evolution of a chain of spin 1/2 particles. They consider different types of bulk unitary gates, in particular cases with and without interactions. They then proceed to solve the reflection equation in these different cases and see how they fit into a 2 layer brickwork pattern and see if they factorise.
The paper is very clearly written and gives a nice introduction to all the concepts that are needed to understand the results. I think the results are relevant and could potentially describe some interesting physical models. I recommend this paper for publication.
Requested changes
- Can the authors comment on they expect the models that they find to be exactly solvable by some sort of Bethe ansatz?
- There is a case between periodic boundary conditions and open boundary conditions. These are quasi-periodic or twisted boundary conditions. In this case you need a matrix G such that [R,GG]=0. Then you can insert G into the transfer matrix and you find that G describes the relation between the last and first site. Can this case be obtained from the results in the paper or would it require a separate analysis. Would this case be interesting?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)