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Quantum Algebraic Geometry Codes

Zhengyi Han

Published Dec 1, 2024• 0 citations
OpenAlex

Abstract

Quantum error correction is an essential aspect of quantum information theory, providing protection for quantum states against noise and decoherence. This thesis investigates the construction of quantum error correction codes derived from classical algebraic geometry (AG) codes. We present two distinct construction techniques, highlighting the flexibility and self-orthogonality of AG codes, and demonstrate their ability to produce asymptotically good quantum codes. Additionally, we explore strategies to fine-tune the parameters of classical AG codes, ensuring they possess the desired properties for quantum code construction. This work serves as a comprehensive guide to the fundamental concepts and common methodologies underlying quantum algebraic geometry codes.

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Quantum error correction is an essential aspect of quantum information theory, providing protection for quantum states against noise and decoherence. This thesis investigates the construction of quantum error correction codes derived from classical algebraic geometry (AG) codes. We present two distinct construction techniques, highlighting the flexibility and self-orthogonality of AG codes, and demonstrate their ability to produce asymptotically good quantum codes. Additionally, we explore strategies to fine-tune the parameters of classical AG codes, ensuring they possess the desired properties for quantum code construction. This work serves as a comprehensive guide to the fundamental concepts and common methodologies underlying quantum algebraic geometry codes.
Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.
Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct two families of QUENTA codes, where one of them has maximal entanglement and is maximal distance separable. In the end, we show that for any asymptotically good tower of algebraic function fields there is an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative amount of entanglement.
In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.
Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct three families of QUENTA codes, where one of them has maximal entanglement and is maximal distance separable. At the end, we show that for any asymptotically good tower of algebraic function fields there is an asymptotically good family of maximal entanglement QUENTA codes with nonzero rate, relative minimal distance, and relative amount of entanglement
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.