Quantum Weight Reduction with Layer Codes

TL;DR

This work introduces a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead.

Abstract

Quantum weight reduction procedures ease the implementation of quantum codes by sparsifying them, resulting in low-weight checks and low-degree qubits. However, to date, only few quantum weight reduction methods have been explored. In this work we introduce a simple and general procedure for quantum weight reduction that achieves check weight 6 and total qubit degree 6, lower than existing procedures at the cost of a potentially larger qubit overhead. Our quantum weight reduction procedure replaces each qubit and check in an arbitrary Calderbank-Shor-Steane code with an ample patch of surface code, these patches are then joined together to form a geometrically nonlocal Layer Code. This is a quantum analog of the simple classical weight reduction procedure where each bit and check is replaced by a repetition code. Due to the simplicity of our weight reduction procedure, bounds on the weight and degree of the resulting code follow directly from the Layer Code construction and hence are easily verified by inspection. Our procedure is well suited for implementation in modular architectures that consist of surface code patches networked via long-range interconnects.

Figures (7)

• Figure 1: (a) The grey circles denote qubits. The blue boxes denote $X$-checks, with lines denoting which qubits the check acts on, and similarly for red boxes, which denote $Z$-checks. (b) $X,Z$-checks and qubits are replaced with surface codes. Here, solid lines depict smooth boundaries, and dashed lines depict rough boundaries.
• Figure 2: Interactions. $X$- (blue) and $Z$-check layers (red) are glued to the corresponding qubit layers by topological defects, and to one another along green string defects that ensure the output is a valid stabilizer code, i.e., all checks commute. The chosen graph coloring ensures the length $\ell$ of the green string defects are consistent. Extended boundary conditions are used for the $X$- and $Z$-check layers williamson2025partial. See Figure \ref{['fig:shor-dimensions']} for the final result for the Shor code.
• Figure 3: Dimensions of the weight reduction layers. Blue, red, and green lines denote topological defects along which pairs of layers are glued, see Fig. \ref{['fig:shor-defect']}. Sizes of the layers are determined by the chromatic numbers of induced graphs. In this example, $\chi_X=\chi_Z=2,$$\chi_Q=6$. For simplicity of presentation, the $\chi_Q$ direction is not drawn to scale.
• Figure 4: Layer Code Example. (a) denotes the Euclidean layer code $C$ constructed from input code $A$ with parity checks $XXI,ZZZ$, where the squiggly line denotes a possible logical representation (b) illustrates the isomorphism in Theorem \ref{['thm:unified']} using the logical representation in (a) as an example. In particular, it corresponds to the logical $XIX$ of the input code $A$.
• Figure 5: Thickening. (a) depicts a cell complex $G$ corresponding to a graph with generating simple cycles $\sF$ colored. (b) depicts the thickened $C$ for $L=3$, where the original cycles $f\in \sF$ are mapped to different heights via $h$. Note that in (a), there exist an edge adjacent to both red and green plaquettes, while any edge in (b) can only be adjacent to either the red or the green plaquette, and thus the weight is reduced by a clever choice of the height function.

Theorems & Definitions (51)

• Theorem 1.1: Main result, corollary of Theorem \ref{['thm:main']}
• Definition 2.1: Graph induced by $A$
• Theorem 2.2: Algorithm \ref{['alg:sparsification']}
• Remark 1
• proof
• Definition 3.1
• Definition 3.2: Basis
• Example 3.1
• Lemma 3.3: Repetition Code
• Definition 3.4: CSS Codes