GNarsil: Splitting Stabilizers into Gauges

TL;DR

This paper introduces algorithms to convert CSS quantum codes into subsystem codes with smaller gauge operators, improving error correction and stabilizer measurement efficiency, demonstrated through new code constructions and comparative analysis.

Contribution

The paper presents novel algorithms for generating subsystem codes from CSS codes, including new code constructions and insights into stabilizer splitting challenges.

Findings

Algorithms recover Bacon-Shor code and produce new codes

New [9,1,2,2] rotated surface subsystem code

Subsystem lifted product (SLP) code outperforms stabilizer versions

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.