General Construction of Quantum Error-Correcting Codes from Multiple Classical Codes

TL;DR

This letter proposes a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D, guaranteeing the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions.

Abstract

The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.

Figures (4)

• Figure 1: All four types of quantum codes constructed from three classical codes are presented. A representative example is shown for each type, with the total number of distinct codes of that type indicated by "$\times n$". The first and third rows correspond to $Z$-check and $X$-check blocks, respectively, while the middle row contains all qubit blocks. The upward- and downward-pointing triangles of different colors adjacent to each block indicate that the block is checked by the $Z$- and $X$-check blocks of the corresponding color.
• Figure 2: The unit cells of the four distinct lattice models corresponding to the four constructions in the $D=3$ case shown in Fig. \ref{['3-d-code']}. Black dots represent qubits. $Z$ and $X$ symbols in different colors denote $Z$-checks and $X$-checks, with colors matching those in Fig. \ref{['3-d-code']}. Green and yellow arrows indicate the action of each check by connecting it to the qubits on which it acts.
• Figure 3: The code dimension $k$ and code distance $d$ for four different kinds of codes at $D=3$. Here, the total number of qubits is fixed at $n=144$ for (a) and $n=432$ for (b). For a fixed $n$, various combinations of $(L_1,L_2,L_3)$ are possible, with each combination corresponding to a distinct data point for a given case. The number $(L_1,L_2,L_3)$ aside the data point represents the combination with which the largest $k$ or the largest $d$ is achieved.
• Figure 4: Illustration of the FLIPs and the induced pairing structure between qubits. (a) The indexed $D$-tuples $\alpha$, $\beta$, and $\theta$ are depicted as colored bars where same color indicates same choices of indices at that position. They represent $Z$-stabilizer, qubit and $X$-stabilizer, respectively. $\beta$ is generated from $\alpha$ by applying $\mathcal{F}_1$ and $\mathcal{F}_2$, yielding the green and orange regions where $\beta$ differs from $\alpha$. It can also be generated by applying $\mathcal{F}_3$ and $\mathcal{F}_4$ to the $X$-stabilizer $\theta$. (b) $\beta'$ is constructed from $\alpha$ using $\mathcal{F}_2$ and $\mathcal{F}_4$, or constructed from $\theta$ using $\mathcal{F}_1$ and $\mathcal{F}_3$. The arrow below indicates that panels (a) and (b) are related by interchanging the order of $\mathcal{F}_1$ and $\mathcal{F}_4$. For visual clarity, the colored segments are drawn contiguously, although contiguity is not required in general.