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Reinforcement learning decoding for Hermitian AG quantum stabilizer codes

Vahid Nourozi

Published Oct 6, 2025• 0 citations
Publisher

Abstract

Abstract We provide a corrected, consistent development of classical algebraic geometry (AG) codes and associated nonbinary quantum stabilizer codes built fromthe Hermitian curve $y^q+y=x^{q+1}$ over $\Fqsq$. This resolves earlier inconsistencies around the choice of $m$, point counts, dual indices, and field of the inner product. The curve has genus $g=\frac{q(q-1)}{2}$, $q^3{+}1$ $\Fqsq$-rational points, and a unique point at infinity; setting $D$ to be the sum of all affine $\Fqsq$-points gives length $n=\deg D=q^3$. For one-point divisors $G=rP_\infty$ we give a monomial basis for $\mathcal{L}(G)$, the standard duality formula $C_{\mathcal{L}}(D,rP_\infty)^\perp=C_{\mathcal{L}}\!\big(D,(n+2g-2-r)P_\infty\big)$, and a simple sufficient condition for Hermitian self-orthogonality: $(q+1)r\le n+2g-2$ (equivalently, $r\le q^2-2$).Under this condition, Ashikhmin--Knill yields $q$-ary stabilizer codes $[[n,n-2k,\ge d_H^\perp]]_q$ with $k=\dim C_{\mathcal{L}}(D,rP_\infty)$ and $d_H^\perp\ge r-2g+2$.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).

DOI
10.21203/rs.3.rs-7365794/v1

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This work develops a geometric framework for constructing quantum error-correcting codes from weighted projective and orbifold structures, integrating algebraic geometry, divisor theory, and the CSS stabilizer formalism. Beginning with weighted projective spaces and their associated height and defect structures, the study builds classical AG-codes via evaluation on divisors adapted to orbifold singularities. These classical codes are lifted to quantum codes using self-orthogonality conditions and homological constructions, yielding a class of Quantum Weighted Algebraic Geometric (QWAG) codes. A central contribution is the formulation of a refined Singleton-type bound motivated by orbifold defect terms and effective genus corrections. While the classical quantum Singleton bound is recovered in the smooth case, the orbifold setting suggests additional geometric contributions that may adjust the theoretical distance bound. The refined bound is presented with partial justification under specific geometric hypotheses and framed as a conjectural extension in full generality. The monograph further provides explicit constructions, computational implementations in Sage/Python, and illustrative examples demonstrating how weighted geometry influences code parameters. This work establishes a structured bridge between orbifold geometry and quantum coding theory, outlining both concrete constructions and open problems for further mathematical development.
We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $ c $, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $ c $ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.
In this paper, we introduced a new class of infinitesimal evaluation codes over the dual-number extension $ R = \mathbb{F}_{q^2}+u \mathbb{F}_{q^2}, u^2 = 0, $ obtained by evaluating polynomials at perturbed points $ a_i+ub_i $. This evaluation produces a coupled value–derivative structure through the identity $ f(a_i+ub_i) = f_0(a_i)+u\bigl(b_i f_0'(a_i)+f_1(a_i)\bigr), $ which enriches classical evaluation codes with first-order infinitesimal corrections. We established the Hermitian duality theory for these codes and showed that Hermitian orthogonality over $ R $ decomposes into a residue-layer condition over $ \mathbb{F}_{q^2} $ together with a correction equation involving the infinitesimal parameters. This yields explicit criteria for Hermitian self-orthogonality. Using these criteria, we constructed several families of Hermitian self-orthogonal infinitesimal evaluation codes, including multiplier perturbation, locator perturbation, and subgroup–coset constructions. Via the Gray map and the Hermitian construction, these codes produce new families of $ q $-ary quantum stabilizer codes.
In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characteristic is slightly less than n/2-g with n being the length of the code and g being the genus of the base curve, then it is equivalent to an Euclidean self-orthogonal code. Previously, such results required a strong condition on the existence of a certain differential. We also show a similar result on Hermitian self-orthogonal algebraic geometry codes. As a consequence, we can apply our result to quantum codes and obtain some good quantum codes. In particular, we obtain a q-ary quantum [[q+1,1]]-MDS code for an even power q which is essential for quantum secret sharing.