SSIP: automated surgery with quantum LDPC codes

TL;DR

SSIP introduces a homology-based framework and software for performing CSS-code surgery via pushouts and coequalisers, enabling both external and internal merges on quantum $\text{qLDPC}$ memories. By leveraging tensor and lifted-product constructions over $\mathbb{F}_2$, SSIP translates Tanner-graph reasoning into chain-complex operations and yields concrete distance-preserving merges with low ancillary overhead. Benchmark results on lift-connected surface codes and generalized/bivariate bicycle codes show substantial reductions in qubit overhead (e.g., the $\llbracket 144,12,12\rrbracket$ gross code can support simultaneous logical measurements with about 150 ancillae) relative to prior predictions, while maintaining code distance under merges. The work also outlines limitations (no fault-tolerant circuit synthesis or decoders) and maps clear paths for future improvements, including threshold analysis, broader code families, and potential uses for magic-state injection via surgery.

Abstract

We present Safe Surgery by Identifying Pushouts (SSIP), an open-source lightweight Python package for automating surgery between qubit CSS codes. SSIP is flexible: it is capable of performing both external surgery, that is surgery between two codeblocks, and internal surgery, that is surgery within the same codeblock. Under the hood, it performs linear algebra over $\mathbb{F}_2$ governed by universal constructions in the category of chain complexes. We demonstrate on quantum Low-Density Parity Check (qLDPC) codes, which are not topological codes in general, and are of interest for near-term fault-tolerant quantum computing. Such qLDPC codes include lift-connected surface codes, generalised bicycle codes and bivariate bicycle codes. We show that various logical measurements can be performed cheaply by surgery without sacrificing the high code distance. For example, half of the single-qubit logical measurements in the $Z$ or $X$ basis on the $[[ 144 ,12, 12 ]]$ gross code require only 30 total additional qubits each, assuming the upper bound on distance given by QDistRnd is tight. This is two orders of magnitude lower than the additional qubit count of 1380 initially predicted by Bravyi et al.

Figures (26)

• Figure 1: $\omega_{\rm after} - \omega_{\rm before}$ for merges between weight 3 logicals. Values for $Z$ merges are shown first and values for $X$ merges second, when weight 3 logicals exist.
• Figure 2: Figures of merit for individual $X$ and $Z$ merges between LCS codes.
• Figure 3: Comparison of LCS code individual $Z$-merges to surface codes and Coh. The first row in each box is our homological approach using Algorithm \ref{['alg:ext_merge']}. The second is lattice surgery with surface code patches. The third is a naive application of Coh to LCS codes.
• Figure 4: Figures of merit for parallel $X$ and $Z$ merges between LCS codes.
• Figure 5: Comparison of LCS code parallel $Z$-merges to surface codes and Coh. The first row in each box is our homological approach using Algorithm \ref{['alg:ext_merge']}. The second is lattice surgery with surface code patches. The third is a naive application of Coh to LCS codes.

Theorems & Definitions (10)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Remark 3.1
• Remark 4.1