GNarsil: Splitting Stabilizers into Gauges

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.

8 Pages, 3 Figures

References in corpus (12)

• Improved Simulation of Stabilizer Circuits
• Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
• A Unified and Generalized Approach to Quantum Error Correction
• The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
• Dynamically Generated Logical Qubits
• Floquet codes without parent subsystem codes
• Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead
• On the logical operators of quantum codes
• Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound
• Logical Clifford Synthesis for Stabilizer Codes
• Graphical CSS Code Transformation Using ZX Calculus
• Automated searching for quantum subsystem codes