Continuous Relaxation Decoding

Continuous Relaxation Decoding for CSS Quantum LDPC Codes

This repository presents a comprehensive exploration of Continuous Relaxation Decoding for CSS Quantum LDPC Codes, focusing on critical aspects such as energy landscape, basin geometry, gradient dynamics, and hardware implementation. The work is authored by Yannick Schmitt and is currently in preprint status as of April 2026, accessible via DOI: 10.5281/zenodo.19484007.

Repository Contents

The repository is organized as follows:

paper/
  Continuous Relaxation Decoding for CSS_Quantum LDPC Codes.tex   # full source
  Continuous Relaxation Decoding for CSS_Quantum LDPC Codes.pdf   # compiled paper

script/
  continuous_relaxation_decoder.py   # core library (decoder, code builders, GF(2) tools)
  verification_ContinuousRelaxationDecoder.py   # unified verification script

misc_computation/
  o1_convergence_radius.py   # basin geometry: three nested radii
  o2_momentum_escape.py        # momentum decoder comparison (GD / HB / NAG / Langevin)
  o3_decoder_threshold.py      # DSP measurement, failure mode census
  o4_css_generalisation.py      # λ* universality across CSS code families
  o5_hardware_acceleration.py  # FLOP model, serial/batch/PyTorch throughput
  cd_smart_explorer.py   # causal diamond seed search, 64 augmenting rows, B4 orbits
  cd_smart_explorer_augmentation.py   # level-2 augmentation and [[468,36]] ISD certification

Decoder Overview

The innovative decoder transforms binary error variables into continuous spin variables, employing a squared-spring energy function to determine codeword validity. The mathematical formulation is expressed as:

$$H(v) = \sum_q \frac{\lambda}{4}(v_q^2-1)^2 + \sum_j \frac{(1-P_j)^2}{4}, \quad P_j = \prod_{q \in \text{supp}(j)} v_q$$

This function achieves global minima at valid codewords, ensuring that the decoder operates effectively based on the derived energy landscape.

Key Results

The repository showcases several pivotal findings:

Numerical Verification

Findings include significant analytical results verified to machine precision, such as:

• Critical Points: The linear spring energy lacks fixed points at valid codewords.
• Single-Error Gradient: The behavior of the energy landscape around a single error is precisely defined.
• Exact Hessian: The Hessian matrix characteristics at valid codewords provide insights into stability and performance.

Usage

To reproduce results or conduct further investigations, utilize the verification script and various analysis scripts provided in the script/ and misc_computation/ directories. Each script serves a specific purpose, either validating theoretical claims or exploring geometric and dynamical properties of error corrections.

Dependencies

The project requires the following Python libraries:

numpy
scipy
torch  # optional for hardware acceleration

All computations rely on the core library in continuous_relaxation_decoder.py, ensuring cohesive functionality across the repository.

This repository serves as a resource for researchers and developers involved in quantum error correction and offers valuable insights into the ongoing advancements in this dynamic field.