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This repository contains the full research for a novel approach to decoding Quantum Low-Density Parity-Check (QLDPC) codes via continuous optimization. Moving away from traditional message-passing or combinatorial algorithms, this work introduces a syndrome-agnostic energy landscape—a \"squared-spring\" potential where global minima correspond exactly to valid codewords. The decoder operates as a codeword projector, using gradient dynamics to navigate the continuous relaxation of the binary error space.

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Key Contributions

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  1.  The Energy Landscape: Formalization of the squared-spring Hamiltonian $H(v)$, proving that valid codewords are the only points with zero energy and zero gradient (Theorem 3.4).
    2. \n
  2. Basin Geometry & Convergence: Analytical derivation of the exact Hessian and gradient at valid codewords (Theorems 4.1, 4.3). The work defines three nested radii ($r_{loc}$, $r_{barrier}$, $r_{emp}$) that characterize the decoder’s convergence behavior.
    3. \n
  3. The Girth–Trap Hypothesis: An empirical finding that challenges existing intuition; data demonstrates that higher girth does not necessarily prevent decoder trapping, with girth-8 codes observed to trap at higher rates than girth-4 variants.
    4. \n
  4. Optimal λ Heuristic: Discovery of a universal scaling law $\\lambda^* \\approx (3/16)\\bar{d}$ for the regularization parameter, verified across eight independent CSS code families.
    5. \n
  5. Hardware Acceleration: A high-throughput implementation leveraging PyTorch and CUDA, achieving >86,000 syndromes per second, representing a ~28,000× speedup over serial Python implementations.
    6. \n
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Contents

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Computational Experiments: 

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Manuscript:  LaTeX source and compiled PDF of the paper.

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Testbed Codes
The results are benchmarked on several high-performance QLDPC codes, including:

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