Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability

Abstract: We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-$Z$ gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-$Z$ gate parallelizability. More precisely, on input a good quantum AG code over the alphabet $\mathbb F_q$ with transversal $\mathsf{C}^m\mathsf Z$ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet $\mathbb F_{q^2}$ supporting a transversal $\mathsf{C}^m\mathsf Z$ gate as well as addressable and parallelizable $\mathsf{C}^{m-1}\mathsf Z$ gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.

• The paper introduces a framework that constructs quantum CSS codes via quasi group codes, enabling transversal non-Clifford gates with improved decoding efficiency.
• It leverages abelian function field extensions and module structures to optimize gate addressability and parallelizability for scalable fault-tolerant computation.
• The approach reduces decoding complexity from cubic to quasi-quadratic, offering significant resource overhead improvements for magic state distillation.

Quantum Group Codes for Non-Clifford Logic: Enhanced Decoding, Addressability, and Parallelizability

Overview and Motivation

This paper introduces a unified framework for constructing quantum CSS codes—specifically, quantum group codes—that exhibit highly desirable properties for fault-tolerant quantum computation. The construction is motivated by limitations in prevailing quantum error-correcting code frameworks, particularly regarding the efficient implementation of non-Clifford gates at the logical level, the parallelizability and addressability of logical gates, and the computational overheads of decoding.

The work builds on both recent advances in algebraic geometry code-based quantum codes (especially those with non-Clifford transversal gate capabilities) and on techniques in classical coding theory pertaining to group and quasi-group module structures. A central outcome is the definition of a broad class of quantum CSS codes, derived via lifting procedures from classical AG codes through class field theory, that attain improved decoding complexity, addressable transversal multi-control-$Z$ logical gates, and extensive gate parallelizability.

Framework: Quasi Group Codes and Quantum CSS Construction

The authors generalize previous approaches by leveraging quasi group codes as the classical basis from which quantum codes are synthesized. By employing classical AG codes structured as quasi modules over large abelian groups, the resulting quantum CSS codes inherit favorable group symmetries. This module structure is critical for enabling the construction of transversal gates that are addressable—meaning that a large fraction of arbitrary subsets of logical qudits can be targeted by transversal non-Clifford gates—and for ensuring parallelizability, i.e., the ability to execute large classes of logical gates in constant or low circuit depth.

Key technical contributions include:

• A precise description of how lifting AG codes through abelian function field extensions (constructed using explicit class field theory techniques [CouGas26]) both preserves the requisite "multiplication properties" for transversal implementation of multi-control-$Z$ gates and endows the codes with a free module structure conducive to efficient parallelization.
• A demonstration that such lifted codes possess an automorphism group containing a large, freely acting abelian subgroup, which governs the code's orbits and supports explicit control over addressability/parallelizability trade-offs.
• Rigorous analysis showing that the duals of these codes maintain the necessary structure, which is crucial for CSS code construction and for satisfying quantum error correction conditions.

Gate Addressability and Parallelizability

The addressability property is formalized via the action of the large abelian group: for each orbit of the group action, subsets of logical qudits can be independently targeted by transversal multi-control-$Z$ gates. Two operational scenarios are constructed:

1. Full Addressability: By puncturing/shortening the code on a single group orbit, the code achieves the maximal degree of logical gate addressability—any $m$-tuple of logical qudits from a single code block can be independently manipulated with a transversal $\mathsf{C}^{m-1}\mathsf{Z}$ gate, and such gates over all $k$ logical qudits can be executed in depth $O(k^{m-1})$.
2. Trade-off Regime: By puncturing multiple orbits, codes with linear rate and distance are constructed wherein a global transversal $\mathsf{C}^m\mathsf{Z}$ gate persists, and orbit-wise addressable gates are available to subsets of the logical qudits. The number and size of orbits explicitly control this trade-off.

These constructions generalize, extend, and unify addressable gate frameworks developed in previous works [he2025good, guemard2025good].

Enhanced Decoding Algorithms

A significant computational advancement is the reduction of the decoding complexity for the constructed code families. The classical AG decoder, when applied to codes with large group module structure, can be accelerated from cubic to quasi-quadratic time in the code length using fast Fourier transform and module decomposition algorithms developed in [CouGas26]. This represents a nearly linear factor speedup compared to earlier AG-code-based quantum error-correcting codes used in state-of-the-art magic state distillation protocols [Wills2025].

Explicitly, for code length $n$, the randomized decoder operates in $O(n^2 \cdot \operatorname{polylog}(n))$ operations and achieves a linear decoding radius, surpassing the previous $Z$0 bounds.

Quantitative Results

Formally, the main theorem establishes (parameterized by alphabet $Z$1 for $Z$2):

• The existence of sequences of quantum group codes over $Z$3 with asymptotic parameters
  • $Z$4 with full addressability of transversal $Z$5 gates for all $Z$6, and depth $Z$7 logical multi-control-$Z$8 circuits for $Z$9 logical qudits.
  • $Z$0 realizing a transversal global $Z$1 gate as well as orbit-wise addressable $Z$2 gates across $Z$3 orbits of size $Z$4.
• In both cases, both $Z$5- and $Z$6-type decoders are available with quasi-quadratic complexity and linear decoding radii.

For magic state distillation, an important practical corollary is provided:

• Existence of a $Z$7 quantum code supporting a constant-overhead distillation protocol for $Z$8-magic states (i.e., with overhead exponent $Z$9), where the time complexity reduces to $m$0, providing significant improvements over prior protocols [Wills2025].

Implications and Prospects

Practical

The reduction in decoding complexity and increase in addressability and parallelizability have direct implications for the overall resource overhead of quantum fault-tolerance. In particular, the presented constructions enable substantially more efficient magic state distillation, which is critical for scaling universal fault-tolerant quantum computers. The high degree of gate addressability unlocks new potential for efficient logical circuit compilation, especially for non-Clifford-dominated quantum algorithms, and improves prospects for reduced circuit depth in large-scale logical operations.

Theoretical

The synthesis of class field theoretic techniques, module-theoretic perspectives, and algebraic coding theory sets a powerful general foundation for future code design. It reveals deep connections between the algebraic structure of the underlying codes and fault-tolerant logical gate sets in the Clifford hierarchy. The modular lifting technique could be extensible to broader classes of classical codes, and potentially to codes over non-abelian group modules via further generalization.

Combination of CSS code techniques with automorphism group exploitation opens further possibilities for jointly optimizing code rate, distance, transversal gate set, and decoder complexity.

Future Developments

Potential avenues include:

• Generalization of the lifting and module structure techniques to accommodate further gates in the Clifford hierarchy, or gates useful for specific quantum algorithms.
• Exploration of quantum LDPC codes incorporating similar group module symmetries to bridge asymptotic rate/distance with hardware-favorable locality.
• Investigation of trade-offs between addressability, code rate, and decoding complexity for application-specific quantum architectures or near-term intermediate-scale quantum (NISQ) devices.
• Integration with hardware-oriented constraints, such as code locality or connectivity, to develop more physically realizable fault-tolerant protocols.

Conclusion

This paper provides a formally rigorous, highly structured construction for quantum CSS codes with enhanced logical gate capabilities and computational efficiency. By leveraging group module symmetries and explicit function field extensions, the authors achieve both improved decoding complexity and maximal logical gate addressability and parallelizability. The resulting framework not only advances state-of-the-art for non-Clifford gate implementation and magic state distillation, but also sets a versatile foundation for the continued evolution of quantum code theory towards scalable, resource-efficient fault-tolerant quantum computation.