Continuous Relaxation Decoding provides innovative solutions for CSS Quantum LDPC Codes by exploring energy landscapes, basin geometry, and gradient dynamics. This repository includes essential resources such as code, papers, and verification scripts, offering valuable insights and tools for researchers and practitioners in quantum error correction.
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:
| Result | Value |
|---|---|
| Primary testbed | [[193, 25, d=4]] D4 causal diamond HGP code |
| Secondary testbeds | [[112, 4, (6,6)]] and [[176, 32, (3,6)]] augmented-seed codes |
| Provably perfect decoding | t ≤ 1 (monotone path theorem); t = 2 empirically |
| Pseudo-codeword barrier heights | ΔE ∈ [1.0, 4.0] at λ = 0.5 |
| Optimal λ heuristic | λ* ≈ (3/16)d̄, confirmed on 6 of 8 code families |
| Girth–trap hypothesis | Falsified: girth-8 code traps at 2.5× the rate of girth-4 |
| GPU throughput | 86,535 syn/s at batch 16,384 (~28,000× faster than serial Python) |
| FLOP vs MWPM | 8× fewer than O(N³) bound; 18× more than sparse-Blossom |
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.
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