construct entanglement-assisted codes

This work addresses the challenge of constructing entanglement-assisted quantum error-correcting codes (EAQECCs) for high-dimensional quantum systems in the presence of noisy shared entangled bits (ebits). By generalizing the stabilizer formalism of binary EAQECCs to the qudit case over an arbitrary finite field 𝔽_q, the authors establish a unified construction framework for q-ary EAQECCs that explicitly accounts for ebit noise. Leveraging the generalized Pauli group over 𝔽_q, symplectic geometry, and additive code theory in 𝔽_q^{2n}, this approach systematically generates various families of q-ary EAQECCs. Notably, the resulting codes outperform optimal standard stabilizer codes with equivalent error-correcting capabilities, thereby demonstrating a clear advantage of EAQECCs in specific noisy environments where pre-shared entanglement is imperfect.

The existence and explicit construction of entanglement-assisted quantum MDS (EAQMDS) codes—a long-standing open problem—hinder the development of optimal entanglement-assisted quantum error-correcting codes (EAQECCs) for reliable, efficient quantum information processing. Method: We introduce, for the first time, a symplectic group geometric approach to systematically establish a structural correspondence between symplectic subspaces and quaternary additive codes, thereby revealing the geometric nature of EA-stabilizer code parameters. Integrating finite-field additive coding theory with the EA-stabilizer formalism, we develop a unified framework for constructing EAQECCs and design low-complexity quantum encoding/decoding circuits. Contribution/Results: Our work yields multiple families of optimal EAQECCs and the first infinite family of explicitly constructed EAQMDS codes, providing both theoretical foundations and practical implementations for high-efficiency, fault-tolerant entanglement-assisted quantum computation.

This work addresses the loose lower bound on the Hermitian hull dimension of generalized rational algebraic geometry codes, which hinders the efficient construction of entanglement-assisted quantum error-correcting codes (EAQECCs). We propose a novel method integrating Weil differential residue properties with the Hermitian dual, enabling—for the first time—explicit lower-bound estimation and precise control of the Hermitian hull dimension. Leveraging algebraic function field theory and generalized Reed–Solomon code constructions, we systematically design two families of maximum-distance-separable (MDS) EAQECCs with unprecedented parameters: their lengths, dimensions, and required numbers of pre-shared ebits all surpass current records. The key innovation lies in establishing an analytic connection between the Hermitian hull dimension and the structure of Weil differentials, thereby providing a scalable algebraic-geometric framework for efficiently constructing high-performance EAQECCs.

Existing constructions of binary quantum error-correcting codes are hindered by the algebraic complexity of enforcing self-orthogonality under the Hermitian inner product, limiting improvements to the minimum distance lower bounds in Grassl’s code table. Method: We establish, for the first time, a sufficient algebraic condition for Hermitian self-orthogonality of binary quasi-cyclic codes with two generators, and leverage it to devise an efficient, algebraically implementable framework for quantum code construction. Contribution/Results: Our approach overcomes the computational bottleneck inherent in conventional self-orthogonality constraints. It yields 30 new binary quantum codes whose minimum distances strictly exceed the previous best-known values in Grassl’s table for the same parameters—thereby significantly advancing the performance frontier for quantum codes of small lengths.

This work addresses the challenge of designing high-rate, low-overhead CSS quantum error-correcting codes. We introduce *dihedral quantum codes*—a novel family of CSS codes constructed via the lifted product—and present the first systematic framework for short-length instances. Leveraging the group algebra structure and representation theory of the dihedral group, we derive an explicit analytical formula for the code dimension in terms of two classical constituent codes and rigorously establish a nontrivial lower bound on the quantum distance. Beyond theoretical development, we provide concrete constructions that empirically validate the explicit trade-off between dimension (i.e., encoding rate) and distance (i.e., error-correction capability). Our results unify structural insights from group theory and coding theory, yielding a new design paradigm for practical quantum codes that simultaneously achieve high rate and robust fault tolerance.

This study investigates the construction of high-performance quantum error-correcting (QEC) codes and entanglement-assisted quantum error-correcting (EAQEC) codes from cyclic codes over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$. By analyzing for the first time the Hermitian hull and Hermitian sum structures of cyclic codes over this ring, the work integrates quantum Construction X, matrix-product codes, and linear complementary dual (LCD) code techniques to propose novel constructions of QEC and EAQEC codes. This approach not only extends the design framework for EAQEC codes but also yields several new families of quantum codes, thereby enriching the theoretical foundation of quantum error correction based on non-traditional algebraic structures.

This work investigates the construction and performance limits of entanglement-assisted quantum locally recoverable codes (EA-qLRCs) without requiring the dual-containing condition. By leveraging pairs of classical locally recoverable codes and employing a CSS-like stabilizer framework, the authors introduce local recovery channels and incorporate hull dimension analysis to establish a unified upper-bound framework on code parameters. A general construction criterion is proposed that dispenses with the dual-containing requirement, yielding a tight Singleton-like bound together with necessary and sufficient conditions for its achievability. The study proves that cyclic codes can attain optimal EA-qLRCs, whereas Tamo–Barg codes achieve optimality only in degenerate cases. Furthermore, two Gilbert–Varshamov-type achievability bounds are derived for $q > 3$, fully characterizing the feasible region of rate, distance, and locality.

This work addresses the systematic construction of quantum error-correcting codes with enhanced Hermitian dual distance and controllable Hermitian hull dimension by introducing a novel approach based on generalized extended codes \( C(u,a) \). By analyzing the position of the vector \( u \) relative to \( C + C^{\perp_H} \) and its interaction with minimum-weight dual codewords, the authors establish, for the first time, sufficient conditions that simultaneously increase both the Hermitian hull dimension and the dual distance. They further provide a direct construction linking such codes to entanglement-assisted quantum codes. Leveraging algebraic tools including Hermitian duality theory, codeword weight analysis, and monomial equivalence, they construct 267 new entanglement-assisted qubit codes (with \( n \leq 40 \)) and 14 qutrit codes (with \( n \leq 25 \)) surpassing entries in Grassl’s table and recent literature, among which 236 qubit and 8 qutrit codes achieve strictly improved parameters.

This work addresses the excessive number of CNOT gates in encoders for entanglement-assisted (EA) quantum QC-LDPC codes by formulating encoder optimization as a row-operation search problem over GF(2), building upon the Sharma–Kumar–Garani (SKG) construction. The authors propose, for the first time, an optimization algorithm that integrates beam search with a Hamming-distance-based heuristic. By decomposing binary matrices derived from CNOT subsequences, the method significantly reduces gate count while preserving the structured nature of the encoder. Evaluated across multiple families of EA quantum QC-LDPC codes, the approach achieves a 7.3%–34.0% reduction in CNOT gate count compared to the SKG baseline and outperforms the Patel–Markov–Hayes synthesis method. Correctness of the optimized encoders is verified through stabilizer table simulations.

This work addresses the lack of structural characterization in traditional approximate quantum error correction (AQEC) for handling diverse noise types. By formulating AQEC within an error-set model, the authors introduce a constrained linear structure to uniformly describe correctable channel families and define new code parameters quantifying worst-case and average recovery performance. For the first time, key structural properties from exact QEC—such as code distance, erasure equivalence, and asymptotically good codes—are extended to the approximate setting, yielding a hierarchical framework that aligns errors with appropriate metrics. Combining the Bény–Oreshkov worst-case and Petz average recovery approaches, and leveraging spectral constraints on linear combinations of Kraus operators alongside partition-based constructions, the authors construct the first asymptotically good AQEC code family applicable to fermionic systems, one-dimensional Rydberg-blockade systems, and deletion errors, with straightforward generalization to multiple physical platforms.