Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes

TL;DR

This work analyzes Hastings's weight-reduction techniques for quantum LDPC codes through the lens of effective distance under single-ancilla syndrome extraction. By constructing tailored single-ancilla measurement schedules for each weight-reduction step—copying, gauging, thickening/choosing heights, and coning—the authors show that the effective distance is largely preserved for many resulting codes, with explicit bounds and corollaries for higher-dimensional HGP codes. Key results include exact preservation of X-distance under copying, favorable bounds for Z-distance, and generalized preservation results for thickened, coned, and generalized thickened codes, extending to higher-dimensional HGP codes. The findings strengthen the case for weight-reduced qLDPC codes in fault-tolerant quantum computation and highlight the robustness of HGP codes to hook errors under practical measurement schemes. Overall, the paper provides a rigorous, multi-technique framework for maintaining fault tolerance while reducing stabilizer weights, with practical implications for near-term and future quantum architectures.

Abstract

Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.

Figures (6)

• Figure 1: Examples for copying and gauging. (a) The qubits $q_1$ lies in the support of 4 different $X$ stabilizer generators $s_{X_1}, \ldots, s_{X_4}$. (b) After performing copying, we obtain 4 copies of $q_1$ which are denoted as $q_{1, 1} \ldots, q_{1,4}$. These copied qubits are connected by 3 new $X$ stabilizer generators in a repetition code-like structure. These copied qubits also lie in the same support of the $Z$ stabilizer generators $s_{Z_1}$ and $s_{Z_2}$. (c) Before gauging, the $s_{X_1}$ has support on 4 different qubits $q_1, \ldots, q_4$. (d) After gauging, we split the $X$ stabilizer generator into 4 copied $X$ stabilizer generators $s_{X_{1,1}}, \ldots, s_{X_{1,4}}$ which are connected by 3 new qubits $q_5, q_6, q_7$ in a repetition code-like structure. The $Z$ stabilizer generators $s_{Z_1}$ and $s_{Z_2}$ are also updated to have support on the new qubits to ensure that they commute with the copied $X$ stabilizer generators.
• Figure 2: As per convention, vertices represent $X$ stabilizer generators, edges are qubits, and highlighted faces are $Z$ stabilizers. In (a) we have a surface code with $d_X = 2$ and $d_Z = 3$. (b) is the surface code but after thickening, i.e., $d_X' = 6$ and $d_Z' = 3$. The red highlighted faces correspond to $Z$ stabilizer generators of the original code ($A_2 \otimes B_0$) and the yellow highlighted faces correspond to $Z$ stabilizer generators in $A_1 \otimes B_1$. (c) is the thickened code after choosing heights. For visibility, we do not highlight the stabilizers in $A_1 \otimes B_1$ as they remain unchanged from thickening. We also note that $q_Z$ does not decrease in this particular example because it was already less than or equal to 3. This example is mainly chosen to illustrate the thickening and height-choosing process.
• Figure 3: A visualization of the coning operation. In (A), we start with a high weight $Z$ stabilizer generator where edges correspond to qubits, faces correspond to $Z$ stabilizer generators, and vertices correspond to $X$ stabilizer generators as per convention. Then in (B), we build a new chain complex that has chains that correspond to the qubits and $X$ stabilizers that neighbor the high weight $Z$ stabilizer generator. Next, in (C), we cone the high weight $Z$ stabilizer generator to obtain the cone code. Finally, in (D), we cellulate the coned cone to obtain the reduced cone code. This figure is obtained from Ref. hastings2021quantum.
• Figure 4: An example of a single-ancilla stabilizer measurement circuit. This circuit measures an $X$ stabilizer generator that acts on five data qubits of some quantum code. In the execution of this stabilizer measurement circuit, an $X$ error happens after the second entangling gate, resulting in bit-flip errors to propagate to the data qubits with indices 3, 4, and 5. The final error after the $Z$ stabilizer measurement can be equivalent to $X_1X_2$ since $X_1X_2X_3X_4X_5$ is in the stabilizer group $\mathcal{S}$ of the quantum code. In this example, we observe that a single elementary fault can result in the propagation of errors to multiple data qubits.
• Figure 5: Schematic for the generalized thickened quantum code. There are $n_c$ copies of the quantum code $\mathcal{Q}$ arranged in $n_c$ columns in the three blocks $Z[T], A$, and $X$. For each of the $n_c$ columns, we have the $Z[T]$ stabilizer generators and $X$ stabilizer generators belonging to that column acting on the qubits in the same column. Similarly, there are $n$ copies of the classical code $\mathcal{C}$ arranged in $n$ rows in the two blocks $Z[B]$ and $A$. For each of the $n$ rows, we have the $Z[B]$ stabilizer generators belonging to that row acting on the qubits in the same row which is reminiscent to how the checks of the classical code act on the classical bits. For the qubits in the region $B$, we have the $Z[B]$ stabilizer generators and $X$ stabilizer generators acting on the qubits in the same column and row respectively.

Theorems & Definitions (44)

• theorem 1: Informal Statement of Effective Distance Preservation for Copied and Gauged Codes
• theorem 2: Informal Statement of Effective Distance Preservation for Thickened and Height-Chosen Codes
• theorem 3: Informal Statement of Distance-Balanced Codes
• theorem 4: Informal Statement of Effective Distance Preservation for Coned Codes
• theorem 5: Informal Statement of Hook Error Absence for Higher Dimensional HGP Codes
• lemma 6: hastings2021quantum
• lemma 7: hastings2021quantum
• remark 8
• theorem 9: zeng2019higher
• lemma 10