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Errorcorrecting codes for fermionic quantum simulation
by YuAn Chen, Alexey V. Gorshkov, Yijia Xu
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Submission summary
Authors (as registered SciPost users):  YuAn Chen · Yijia Xu 
Submission information  

Preprint Link:  https://arxiv.org/abs/2210.08411v4 (pdf) 
Date submitted:  20231128 18:46 
Submitted by:  Xu, Yijia 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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 twodimensional 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 $\lfloor \frac{d1}{2} \rfloor$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 syndromematching 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 20231227 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2210.08411v4, delivered 20231227, doi: 10.21468/SciPost.Report.8347
Strengths
1Show how to systematically generate higher distance fermion to qubit encodings using very elegant mathematical framework
2 Clear and pedagogical presentation
Weaknesses
1No discussion of quantum circuit level implementation of the codes.
2Only the most naive, minimum weight, decoding is mentioned.
Report
The manuscript provides a systematic approach to generating larger code distance encodings of fermions in qubit systems. This is presented elegantly using the Laurent polynomial representation of Pauli operators and automorphisms of the symplectic form. The authors manage to present the quite abstract material in an accessible way , with plenty of illustrative examples and figures. The paper may have benefitted from an extended discussion of how the larger code distance encodings would be implemented in a quantum computer. Even under an assumption of perfect stabilizer measurements, the decoding seems nontrivial given the large codespace. It would also be interesting to see a discussion of how the larger code distance encodings map standard fermion problems, such as the Hubbard model. Nevertheless, I think these are topics that can be left for future study. I recommend that the paper should be accepted to SciPost.
Requested changes
No further revision required
Report #1 by Anonymous (Referee 2) on 20231211 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:2210.08411v4, delivered 20231211, doi: 10.21468/SciPost.Report.8241
Strengths
1). The authors provide a systematic way of increasing the fermionqubit code distances while fixing the code rate.
2). The presentation of the background and method is clear.
Weaknesses
1). The discussion of the significance/implications of their findings is limited.
Report
In this paper, the authors introduced a systematic method (Laurent polynomials) to generalize the previously proposed 2D bosonization to a fermion code with a larger code distance while preserving the code rate.
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 higherdistance 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 faulttolerant fermionic quantum simulation.