Native Non-Clifford Gates Achieved with 2D Product Codes and Non-Abelian qLDPC

The challenge of balancing connectivity and universality remains a central problem in fault-tolerant quantum computing with low-density parity-check (qLDPC) codes. Guanyu Zhu of IBM Quantum, alongside Ryohei Kobayashi from the Institute for Advanced Study and Po-Shen Hsin of King’s College London, and their colleagues present a significant advance in addressing this issue. Their research extends Kitaev’s framework for non-Abelian codes to non-Abelian qLDPC codes, utilising topological quantum field theory (TQFT) and innovative gauging measurements. This work demonstrates the possibility of performing native logical non-Clifford gates using two-dimensional product codes, a feat previously thought to require three-dimensional constructions. By effectively implementing addressable gauging measurements, the authors pave the way for more efficient and scalable quantum computation, exemplified by their application to the magic state fountain scheme for parallel preparation of crucial quantum states.

By effectively implementing addressable gauging measurements, the authors pave the way for more efficient and scalable quantum computation, exemplified by their application to the magic state fountain scheme for parallel preparation of crucial quantum states.

Scientists Method

Scientists addressed a critical challenge in fault-tolerant quantum computing , the balance between code connectivity and universality , by extending Kitaev’s framework of non-Abelian codes to encompass non-Abelian quantum low-density parity-check (qLDPC) codes. The research pioneered a method to represent these codes as Clifford-stabilizer codes, linked to combinatorial field theories known as topological quantum field theories (TQFTs), defined on Poincaré CW complexes and general chain complexes.

This innovative approach allowed the team to construct spacetime path integrals as invariants on these complexes, providing a novel mathematical foundation for code design. Remarkably, the study demonstrated that native non-Clifford logical gates could be realised using constant-rate 2D hypergraph-product codes and their Clifford-stabilizer variants, circumventing the need for traditionally proposed 3D constructions. This was achieved through a spacetime path integral that effectively implements addressable gauging measurement of 0-form subcomplex symmetries, which correspond to addressable transversal Clifford gates.

When lifted to higher-dimensional complexes, these symmetries become higher-form symmetries, enabling a powerful new mechanism for gate implementation. Building on this structure, the researchers applied the gauging protocol to the magic state fountain scheme, successfully preparing disjoint CZ magic states with a code distance of 9 using a total of qubits. The team engineered a system to evaluate the cup product on both the skeleton chain complex and the CW complex, establishing a crucial link between the code parameters and the underlying mathematical structure.

This precise evaluation facilitated the development of twisted skeleton hypergraph-product codes, offering a scalable approach to quantum error correction. Furthermore, the work details the construction of Clifford stabilizer models by gauging a higher-form symmetry protected topological (SPT) phase, demonstrating a pathway to create twisted qLDPC codes. The study also explored the non-Abelian fusion rules and Borromean ring braiding, utilising both path integral and operator-based derivations to characterise the complex behaviour of these codes and their potential for universal quantum computation.

This methodological innovation unlocks the possibility of achieving universal fault-tolerant quantum computation with reduced dimensionality and connectivity requirements.

Hypergraph Codes Enable Transversal Clifford Gates

Scientists have achieved a breakthrough in quantum error correction, demonstrating that native non-Clifford logical gates can be realised using constant-rate 2D hypergraph-product codes. This work extends Kitaev’s framework of non-Abelian codes to non-Abelian qLDPC codes, realised as Clifford-stabiliser codes, and defines corresponding combinatorial field theories on Poincaré CW complexes and general chain complexes.

The team constructed spacetime path integrals as invariants on these complexes, revealing a pathway to perform logical operations without requiring three-dimensional constructions previously thought necessary. Experiments revealed the successful implementation of a new type of 0-form subcomplex symmetry, effectively enabling addressable transversal Clifford gates. These symmetries become higher-form symmetries when lifted to higher-dimensional CW complexes or manifolds, providing a powerful mechanism for manipulating quantum information.

Building on this structure, researchers applied a gauging protocol to the magic state fountain scheme, achieving parallel preparation of O(√n) disjoint CZ magic states with a code distance of O(√n), utilising a total of n qubits. Measurements confirm the encoding rate and distance characteristics of the twisted hypergraph-product code, demonstrating its efficiency in representing quantum information. The study details how the cup product, a key element in the construction, can be explicitly evaluated on both the skeleton chain complex and the CW complex, validating the theoretical framework.

Tests prove the viability of gauging measurement of the 0-form subcomplex symmetries, which are crucial for implementing logical non-Clifford operations. The research further explores the non-Abelian fusion rules and braiding statistics inherent in these codes, demonstrating a Borromean ring braiding path integral and an operator-based derivation. This breakthrough delivers a significant advancement in the field of quantum computation, potentially reducing the spacetime overhead required for universal fault-tolerant quantum computing and offering a pathway towards more efficient and scalable quantum architectures.

The work establishes a foundation for future investigations into the interplay between non-Abelian codes, qLDPC codes, and topological quantum field theories.

Gauging Hypergraph Products For Magic State Creation

This work introduces a novel approach to constructing non-Abelian quantum low-density parity-check (qLDPC) codes, extending Kitaev’s framework to encompass Clifford-stabilizer codes and associated topological quantum field theories. Researchers have demonstrated that native non-Clifford logical gates can be realised using constant-rate 2D hypergraph-product codes, a significant departure from previous reliance on three-dimensional constructions.

This achievement stems from a spacetime path integral that effectively implements addressable gauging measurements of specific subcomplex symmetries within the code structure. The authors successfully applied this gauging protocol to the magic state fountain scheme, enabling the parallel preparation of a scalable number of disjoint CZ magic states using a total of n qubits, where the code distance also scales with the square root of n. While acknowledging limitations related to the complexity of evaluating cup products on general chain complexes, the study outlines future research directions focused on exploring the potential of these codes in more complex topological settings.

Further investigation into the precise overheads associated with implementing these gauging measurements and exploring alternative gauging protocols could refine the practicality of this approach to fault-tolerant quantum computation.