SciPost Submission Page
Integrability of open boundary driven quantum circuits
by Chiara Paletta, Tomaž Prosen
Submission summary
| Authors (as registered SciPost users): | Chiara Paletta |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2406.12695v3 (pdf) |
| Date submitted: | 2024-10-10 09:15 |
| Submitted by: | Paletta, Chiara |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| 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
Author comments upon resubmission
Report 1
1. All the models treated in this paper are constructed to be Yang-Baxter integrable, and hence, various integrability techniques (such as the Bethe ansatz or Baxter Q-operators) can be used to solve them. For instance, in certain cases, the algebraic Bethe ansatz can be applied. We refer to Sklyanin's original paper for the case of diagonal K-matrices. Following this comment, we have also added additional references in the manuscript for the cases with non-diagonal boundaries and some examples where Bethe ansatz techniques were applied in the context of quantum circuits.
2. In the context of quantum circuits, the dynamics generated by this type of transfer matrix is reminiscent of the case with open boundary conditions, where the length of the spin chain is even. As mentioned by Sklyanin in the original paper, if a specific representation of the K-matrix is chosen, the double-row transfer matrix reduces to a transfer matrix with quasi-periodic boundary conditions. In our paper, we use a different representation, so this solution is not included in our analysis. However, we found this question interesting and have added Appendix B to clarify the dynamics arising from this case. We have checked that for our models, the twist G also factorizes.
Report 2
No changes or questions asked
Report 3
1. This was indeed an important point. We have expanded the paragraph “Motivation of the paper” at the end of page 4. Additional and more precise motivations are given in section 5.
2. We thank the referee for the comment. We were already aware of this analogy and the relevant references, as it was previously pointed out to us by Z. Bajnok. Based on the referee's suggestion, we have expanded the sentence in the conclusion. However, we did not explore this topic further, though we are open to further discussion if the referee would like to suggest additional cases to investigate.
List of changes
For report 1:
1. We have answered this question at the end of section 2.2.1. To clarify, we also cited cited:
- J. Phys. A: Math. Gen. 21 2375 for Bethe ansatz with diagonal boundary
- 2012.08367 for Bethe ansatz with non-diagonal boundary
- 2408.474 Bethe ansatz for periodic quantum circuits
2. In 2.2 we added the sentence: “For completeness, in Appendix B, we also consider twisted boundary conditions.”
We have added appendix B. To clarify, we added the references: 2207.14193 and 1009.4118
For Report 3:
1. We changed the title of a paragraph in the Introduction: “Integrability in boundary driven diffusive Lindbladian systems”.
We have expanded the paragraph “Motivation of the paper” at the end of page 4.
We added in 3.1 “For the reason that we briefly mentioned in the introduction and that will be clarified in the following,”
2. We have expanded the sentence in the conclusion about IQFT.