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arXiv:2506.03397

Reinforcement Learning Enhanced Greedy Decoding for Quantum Stabilizer Codes over $\mathbb{F}_q$

Vahid Nourozi

Published Jun 3, 2025• 0 citations
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Abstract

We construct new classical Goppa codes and corresponding quantum stabilizer codes from plane curves defined by separated polynomials. In particular, over $\mathbb{F}_3$ with the Hermitian curve $y^3 + y = x^4$, we obtain a ternary code of length 27, dimension 13, distance 4, which yields a [[27, 13, 4]]$_3$ quantum code. To decode, we introduce an RL-on-Greedy algorithm: first apply a standard greedy syndrome decoder, then use a trained Deep Q-Network to correct any residual syndrome. Simulation under a depolarizing noise model shows that RL-on-Greedy dramatically reduces logical failure compared to greedy alone. Our work thus broadens the class of Goppa- and quantum-stabilizer codes from separated-polynomial curves and delivers a learned decoder with near-optimal performance.

DOI
10.48550/arxiv.2506.03397

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Reinforcement Learning Enhanced Greedy Decoding for Quantum Stabilizer Codes over Fq\mathbb{F}_qFq​
We construct new classical Goppa codes and corresponding quantum stabilizer codes from plane curves defined by separated polynomials. In particular, over F3\mathbb{F}_3F3​ with the Hermitian curve y3+y=x4y^3 + y = x^4y3+y=x4, we obtain a ternary code of length 27, dimension 13, distance 4, which yields a [[27, 13, 4]]3_33​ quantum code. To decode, we introduce an RL-on-Greedy algorithm: first apply a standard greedy syndrome decoder, then use a trained Deep Q-Network to correct any residual syndrome. Simulation under a depolarizing noise model shows that RL-on-Greedy dramatically reduces logical failure compared to greedy alone. Our work thus broadens the class of Goppa- and quantum-stabilizer codes from separated-polynomial curves and delivers a learned decoder with near-optimal performance.
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Abstract We provide a corrected, consistent development of classical algebraic geometry (AG) codes and associated nonbinary quantum stabilizer codes built fromthe Hermitian curve yq+y=xq+1y^q+y=x^{q+1}yq+y=xq+1 over $\Fqsq$. This resolves earlier inconsistencies around the choice of mmm, point counts, dual indices, and field of the inner product. The curve has genus g=q(q−1)2g=\frac{q(q-1)}{2}g=2q(q−1)​, q3+1q^3{+}1q3+1 $\Fqsq$-rational points, and a unique point at infinity; setting DDD to be the sum of all affine $\Fqsq$-points gives length n=deg⁡D=q3n=\deg D=q^3n=degD=q3. For one-point divisors G=rP∞G=rP_\inftyG=rP∞​ we give a monomial basis for L(G)\mathcal{L}(G)L(G), the standard duality formula CL(D,rP∞)⊥=CL ⁣(D,(n+2g−2−r)P∞)C_{\mathcal{L}}(D,rP_\infty)^\perp=C_{\mathcal{L}}\!\big(D,(n+2g-2-r)P_\infty\big)CL​(D,rP∞​)⊥=CL​(D,(n+2g−2−r)P∞​), and a simple sufficient condition for Hermitian self-orthogonality: (q+1)r≤n+2g−2(q+1)r\le n+2g-2(q+1)r≤n+2g−2 (equivalently, r≤q2−2r\le q^2-2r≤q2−2).Under this condition, Ashikhmin--Knill yields qqq-ary stabilizer codes [[n,n−2k,≥dH⊥]]q[[n,n-2k,\ge d_H^\perp]]_q[[n,n−2k,≥dH⊥​]]q​ with k=dim⁡CL(D,rP∞)k=\dim C_{\mathcal{L}}(D,rP_\infty)k=dimCL​(D,rP∞​) and dH⊥≥r−2g+2d_H^\perp\ge r-2g+2dH⊥​≥r−2g+2.We also retain a complete reinforcement-learning (RL) section: an \emph{RL-on-Greedy} decoder that first applies a greedy pass then a Deep Q-Network to resolve residual syndromes. We include the original evaluation plot and all references, and clarify that the RL case study can be applied to any stabilizer code (including, but not limited to, those from Hermitian AG codes).
Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers. In order to solve the problem of influence of errors on physical qubits, we propose an approximate error correction scheme that performs dimension mapping operations on surface codes. This error correction scheme utilizes the topological properties of error correction codes to map the surface code dimension to three dimensions. Compared to previous error correction schemes, the present three-dimensional surface code exhibits good scalability due to its higher redundancy and more efficient error correction capabilities. By reducing the number of ancilla qubits required for error correction, this approach achieves savings in measurement space and reduces resource consumption costs. In order to improve the decoding efficiency and solve the problem of the correlation between the surface code stabilizer and the 3D space after dimension mapping, we employ a reinforcement learning (RL) decoder based on deep Q-learning, which enables faster identification of the optimal syndrome and achieves better thresholds through conditional optimization. Compared to the minimum weight perfect matching decoding, the threshold of the RL trained model reaches 0.78%, which is 56% higher and enables large-scale fault-tolerant quantum computation.
This paper characterizes Goppa codes of certain maximal curves over finite fields defined by equations of the form [Formula: see text]. We investigate algebraic geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulation results, which validate the performance of these codes under various error rates. We provide concrete examples of the constructed codes, comparing them with known results to highlight their strengths and trade-offs. The simulation data, presented through detailed graphs and tables, offers insights into the practical behavior of these codes in noisy environments. Our findings demonstrate that while the constructed codes may not always achieve optimal minimum distances, they offer systematic construction methods and interesting parameter trade-offs that could be valuable in specific applications or for further theoretical study.