# Louvre: Relaxing Hardware Requirements of Quantum LDPC Codes by Routing with Expanded Quantum Instruction Set

## TL;DR

Louvre is a routing framework that reduces qubit connectivity requirements in quantum LDPC codes, maintaining performance while enabling more practical implementation on superconducting quantum processors.

## Contribution

This work introduces Louvre, a novel routing-based method that significantly reduces qubit connectivity in quantum LDPC codes without compromising error correction performance.

## Key findings

- Louvre-7 reduces average qubit degree to 4.5 with no performance loss.
- Louvre-8 further reduces degree to 4 with minimal error rate increase.
- Louvre effectively eliminates long-range connections, improving practicality.

## Abstract

Generalized bicycle codes (GB codes) represent a promising family of quantum low-density parity-check codes, characterized by high code rates and relatively local qubit connectivity. A subclass of the GB code called bivariate bicycle codes (BB codes) has garnered significant interest due to their compatibility with two-layer connectivity architectures on superconducting quantum processors. However, one key limitation of BB codes is their high qubit connectivity degree requirements (degree 6), which exacerbates the noise susceptibility of the system. Building on the recent progress in implementing multiple two-qubit gates on a single chip, this work introduces Louvre -- a routing-based framework designed to reduce qubit connectivity requirements in GB codes. Specifically, Louvre-7 achieves degree reduction while preserving the depth of the syndrome extraction circuit, whereas Louvre-8 further minimizes the connectivity by slightly increasing the circuit depth. When applied to BB codes, these two schemes could reduce the average degree to 4.5 and 4, respectively. Crucially, Louvre eliminates some of the long-range, error-prone connections, which is a distinct advantage over prior approaches. Numerical simulations demonstrate that Louvre-7 has an indistinguishable logical error rate as the standard syndrome extraction circuits of GB codes, while Louvre-8 only incurs a slight error rate penalty. Furthermore, by reordering some of the gates in the circuit, we can reduce the coupler length without degrading the performance. Though most of our analysis focuses on GB codes defined on periodic boundary conditions, we further discuss the adaptability of Louvre to open-boundary lattices and defect-containing grids, underscoring its broader applicability in practical quantum error correction architectures.