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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 | |
|---|---|
| Preprint Link: | scipost_202308_00020v1 (pdf) |
| Date submitted: | 2023-08-16 05:15 |
| Submitted by: | Chen, Yu-An |
| 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.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2023-11-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202308_00020v1, delivered 2023-11-10, doi: 10.21468/SciPost.Report.8088
Strengths
1. The authors proposed a way to generate higher distance codes
2. Clear presentation of the methods and algorithms
3. Pedagogical discussion of 2D bosonization, the stabilizer code formalism, and the Pauli module representation via Laurent polynomials.
Weaknesses
1. The authors present a rather straightforward generalisation of previously known models, using standard techniques.
2. There is no discussion of implementation of these models on NISQ hardware.
Report
The authors present a generalisation of 2D bosonization [24] to codes with larger distances. They explain how to use the Laurent polynomials method to perform these extensions in a general way.
This paper should be of interest to experts in the quantum computations/quantum information communities, and I recommend it for publication in SciPost.
Requested changes
I have noticed some spelling errors, which should be corrected.
Author: Yu-An Chen on 2023-11-23 [id 4142]
(in reply to Report 1 on 2023-11-10)We are grateful for the referee's insightful summary and constructive feedback. We have revised our manuscript to address the issues raised.
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.
We now turn to address the concerns highlighted by the referee.
1. The referee noted that our work appears to be a straightforward generalization of existing models employing standard techniques. While we acknowledge that both the models and techniques have been previously established, as mentioned at the bottom of page 3, the innovation of our work lies in the novel application of the "Laurent polynomial" analytical tool to a numerical search algorithm. This synergistic approach could solve other optimization problems in quantum coding.
2. Implementing these models on NISQ hardware is a crucial consideration for the practical application of our proposals. The intricate discussion of implementing our fermion-to-qubit mappings highly depends on the specific hardware platforms and requires in-depth investigation. We intend to explore this in subsequent research. Our current manuscript compares the Pauli weights of standard Hamiltonian terms relative to other established approaches in Table I. This comparison offers insights into the practicality of implementing these mappings in experimental settings.
Anonymous on 2023-11-29 [id 4157]
(in reply to Yu-An Chen on 2023-11-23 [id 4142])I am happy with the response, and recommend the paper for publication.