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Error-correcting codes for fermionic quantum simulation
by Yu-An Chen, Alexey V. Gorshkov, Yijia Xu
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Submission summary
| Authors (as registered SciPost users): | Yu-An Chen · Yijia Xu |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2210.08411v4 (pdf) |
| Date submitted: | Nov. 28, 2023, 6:46 p.m. |
| Submitted by: | Xu, Yijia |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Utilizing the framework of $\mathbb{Z}_2$ lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of $d=2,3,4,5,6,7$, allowing correction of any $\lfloor \frac{d-1}{2} \rfloor$-qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of $d=3,4,5$. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.
List of changes
1. We have proofread the text and equations, correcting any typos present.
2. The description of the symplectic group and automorphisms has been elaborated upon. Section 3.1 has been comprehensively rewritten, with the complete list of generators for the symplectic group added to Appendix B.
3. We have refined the introduction to enhance clarity.
4. Additional references related to the topic have been incorporated into the introduction.
5. The manuscript's format has been updated to conform to the guidelines provided by SciPost Physics.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2023-12-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2210.08411v4, delivered 2023-12-27, doi: 10.21468/SciPost.Report.8347
Strengths
1-Show how to systematically generate higher distance fermion to qubit encodings using very elegant mathematical framework 2- Clear and pedagogical presentation
Weaknesses
2-Only the most naive, minimum weight, decoding is mentioned.
Report
Requested changes
No further revision required
Report #1 by Anonymous (Referee 2) on 2023-12-11 (Contributed Report)
- Cite as: Anonymous, Report on arXiv:2210.08411v4, delivered 2023-12-11, doi: 10.21468/SciPost.Report.8241
Strengths
2). The presentation of the background and method is clear.
Weaknesses
Report
The presentation is clear and their findings are very interesting.
I find their findings useful for the quantum information and computation community, so I recommend their works be published in SciPost Physics.
Requested changes
I have just a few questions/suggestions:
1). Is it possible to represent the Pauli $Y_l$ (or equivalently $X_l Z_l$ with $l$ being a link) operators using the Laurent polynomial vector? It is not very clear to me because $X_l$ and $Z_l$ do not commute.
2). In Section 3, the authors discussed using automorphism ($A$ matrices) to generate higher-distance codes. It would be helpful and more intuitive if they could also present the explicit form of some of the $A$ matrices, namely the corresponding unitary transformations/circuits in terms of the Pauli operators.
3). I find the Discussion section is rather short and it would be great if the authors could elaborate more on connecting their findings to the fault-tolerant fermionic quantum simulation.