GNarsil: Splitting Stabilizers into Gauges

TL;DR

GNarsil introduces two algorithms to derive subsystem codes from a seed CSS stabilizer code by decomposing high-weight stabilizers into products of low-weight gauge operators, while preserving compatible logical operators. The approach yields notable new codes, including the Bacon-Shor $[\[ 9,1,4,3 \\]]$ and a $[\[ 9,1,2,2 \\]]$ rotated surface subsystem code, and demonstrates SHP improvements and a novel SLP construction that can outperform stabilizer counterparts for certain base matrices. However, LP-based codes resist effective gauge decomposition, motivating the SLP framework to achieve competitive parameters (e.g., $[\[ 775,124,20 \\]]$) and higher rates, albeit with memory considerations. The work also discusses non-locality effects on splitting stabilizers and outlines future directions to integrate these subsystem structures with fault-tolerant operations and code symmetries.

Abstract

Quantum subsystem codes have been shown to improve error-correction performance, ease the implementation of logical operations on codes, and make stabilizer measurements easier by decomposing stabilizers into smaller-weight gauge operators. In this paper, we present two algorithms that produce new subsystem codes from a "seed" CSS code. They replace some stabilizers of a given CSS code with smaller-weight gauge operators that split the remaining stabilizers, while being compatible with the logical Pauli operators of the code. The algorithms recover the well-known Bacon-Shor code computationally as well as produce a new $\left[\left[ 9,1,2,2 \right]\right]$ rotated surface subsystem code with weight-$3$ gauges and weight-$4$ stabilizers. We illustrate using a $\left[\left[ 100,25,3 \right]\right]$ subsystem hypergraph product (SHP) code that the algorithms can produce more efficient gauge operators than the closed-form expressions of the SHP construction. However, we observe that the stabilizers of the lifted product quantum LDPC codes are more challenging to split into small-weight gauge operators. Hence, we introduce the subsystem lifted product (SLP) code construction and develop a new $\left[\left[ 775, 124, 20 \right]\right]$ code from Tanner's classical quasi-cyclic LDPC code. The code has high-weight stabilizers but all gauge operators that split stabilizers have weight $5$, except one. In contrast, the LP stabilizer code from Tanner's code has parameters $\left[\left[ 1054, 124, 20 \right]\right]$. This serves as a novel example of new subsystem codes that outperform stabilizer versions of them. Finally, based on our experiments, we share some general insights about non-locality's effects on the performance of splitting stabilizers into small-weight gauges.

Figures (3)

• Figure 1: A diagram describing the relation of the Gauge group $\mathcal{G}$ to the other important subsets of $\mathcal{P}_{n}$ for a subsystem code.
• Figure 2: The Bacon-Shor code and its weight-$2$ gauge operators.
• Figure 3: The rotated surface subsystem code found by Algorithm \ref{['alg:euclid']}. The two red (resp. green) operators are the $X$-gauges (resp. $Z$-gauges), all weight-$3$ (also see Table \ref{['tab:surface_subsystem_gauges']}). The product of the pair of red gauge operators, $X_1 X_2 X_3$ and $X_3 X_4 X_5$, gives $SX_{1}$, and the product of the pair of green operators gives $SZ_{1}$. Stabilizers $SX_{2}$, $SZ_{2}$ can be obtained by multiplying $GX_{1}$, $GZ_{1}$ by the respective stabilizers to find the respective dependent gauges. $GX_{2}$, $GZ_{2}$ are not shown here.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Theorem 3