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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:  (pdf)
Date accepted: 2024-01-09
Date submitted: 2024-01-01 06:02
Submitted by: Chen, Yu-An
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


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.

Author comments upon resubmission

Response to ``Report 2 submitted on 2023-12-27 13:42 by Anonymous'':

Strengths 1-Show how to systematically generate higher distance fermion to qubit encodings using very elegant mathematical framework 2- Clear and pedagogical presentation

Weakness 1-No discussion of quantum circuit level implementation of the codes. 2-Only 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 non-trivial 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.

-We thank the referee for their encouraging feedback and valuable suggestions. In response to the comments, we have enriched our manuscript with an additional paragraph in the discussion section, describing the practical implementation of higher-distance exact bosonization on quantum computers. Furthermore, we have incorporated a paragraph that explores the importance of researching decoders and fault-tolerant gate sets as prospective avenues for future work. Given that exact bosonization originates from the toric code, we anticipate that the minimum weight perfect matching approach still works. With respect to the query about the characteristics of the Hubbard Hamiltonian in the context of higher-distance encoding, our manuscript now includes a detailed description of the $d=3$ construction. Specifically, we elucidate that in this construction, the hopping terms are of weight 3~5, while the interaction terms carry a weight of 6.

Response to ``Anonymous Report 1 on 2023-12-11 (Contributed Report)'':

Strengths 1). The authors provide a systematic way of increasing the fermion-qubit 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.

-We appreciate the referee's insightful feedback. In response, we have revised the discussion section to not only revisit previous analytical applications of Laurent polynomials but also to specifically emphasize our discovery of their effectiveness in numerical calculations. This highlights a significant and practical dimension to their use. Additionally, we have expanded the discussion further to explore future directions in fault-tolerant fermionic quantum simulation.

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.

-We thank the referee for bringing up this point. Yes, since we only care about either two Pauli operators commuting or anti-commuting with each other and the phase factors +-1 and +-i do not affect the commutation relations, here we use X_l Z_l to represent Y_l, which is equivalent to quotient out the phase factors {1,i} from the Pauli group. The detailed arguments can be found in Ref. [55, 56].

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.

-We have drawn a figure to explicitly express the unitary circuit corresponding to each automorphism A1~A16. The figure is attached in Section 3.1.

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.

-We are grateful to the referee for highlighting this aspect. In response, we have incorporated discussions on the essential components needed for fault-tolerant fermionic quantum simulation. We acknowledge that the development of efficient decoders and fault-tolerant gate sets is crucial for these simulations. These topics have been identified as areas for future research and investigation.

List of changes

1. In Section 2.2, we add a couple of paragraphs to discuss the application of d=3 bosonization to the 2d spinless Fermi-Hubbard model.
2. In Section 3.1, we elaborate on the Laurent polynomial formalism, including the representation of the Pauli Y operator.
3. In Section 3.1, we add Fig.6, which provides the unitary circuit descriptions of the sixteen elementary automorphisms used in this work.
4. In Section 4, we elaborate more about the significance and implications of this work. We describe the process of obtaining the codeword of higher-distance encoding from a codeword of the exact bosonization by applying a depth geometrically local and translational-invariant Clifford circuit corresponding to automorphism A. We also discuss the future research directions toward fault-tolerant fermionic quantum simulation.
5. More references for the previous research about Laurent polynomials and quantum codes were added.
6. Some typos are fixed.

Published as SciPost Phys. 16, 033 (2024)

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