Sun and Colleagues Introduce Majorana-Pauli Stabilizer Codes for Fermionic Topological Phases

Meng Sun of Peking University and colleagues have created a new framework for understanding intrinsically fermionic topological phases through Majorana-Pauli stabilizer codes. They present an exactly solvable model of the fermionic toric code, a fundamentally fermionic topological order, by coupling mathbb Z_8 Pauli operators with Majorana modes. This advances the systematic description of fermionic topological phases, revealing connections between bosonic topological orders, symmetry-enriched phases, and symmetry-protected topological phases via a shared stabilizer description. Moreover, the team extends this construction to encompass all Abelian fermionic topological orders with gapped boundaries and establishes an exact bosonization map for mathbb Z’DF symmetries, ultimately bridging fermionic quantum many-body physics and quantum error correction.

Constructing Solvable Fermionic Models via Hybrid Majorana-Pauli Stabilizer Codes Majorana-Pauli stabilizer codes offer a novel approach to constructing exactly solvable models of fermionic systems. These codes combine generalised Pauli operators, mathematical tools for manipulating quantum states and observable properties, with Majorana operators, which describe particles that are their own antiparticles, a characteristic not observed in conventional electrons. Unlike Dirac or Weyl fermions, Majorana modes exhibit unique non-Abelian exchange statistics, making them promising candidates for topological quantum computation. A new type of stabilizer is defined by this hybrid construction, representing a method of encoding quantum information in a manner that protects it from decoherence and errors, analogous to redundancy in computer data storage but leveraging the principles of quantum mechanics. The use of stabilizers ensures that the system remains within a well-defined subspace of the Hilbert space, simplifying analysis and enabling exact solutions. The framework provides a systematic way to analyse intrinsically fermionic topological phases, materials exhibiting complex entanglement patterns previously difficult to model using conventional techniques. These phases are characterised by exotic excitations known as anyons, which possess fractional statistics and are crucial for topological quantum computation. Using these codes, the team constructed a realization of the fermionic toric code, a topological order exhibiting long-range entanglement and protected edge states. This differs significantly from previous bosonic models of the toric code, which rely on spin-like excitations. The fermionic toric code exhibits fundamentally different properties due to the fermionic nature of its underlying constituents. Extending to all Abelian fermionic topological orders with gapped boundaries, meaning the system has a finite energy gap separating the ground state from excited states, the framework avoids reliance on free-fermion analogies or polynomial representations, which often introduce approximations and limit the accuracy of the model. This offers a powerful and versatile tool for exploring these complex systems and understanding their emergent behaviour. The ability to systematically construct solvable models is crucial for verifying theoretical predictions and guiding the search for materials exhibiting these exotic phases. Majorana-Pauli codes solve fermionic topological phases with enhanced algebraic control Stabilizer codes now realise intrinsically fermionic topological orders with a previously unattainable level of algebraic control, as prior methods lacked a simple and complete stabilizer description for these phases. An exactly solvable stabilizer realization of the fermionic toric code, a fundamentally fermionic topological order, was constructed by scientists at Peking University and Stony Brook University, utilising mathbb Z8 Pauli operators coupled to Majorana modes. The mathbb Z8 Pauli operators introduce a more complex algebraic structure compared to standard Pauli matrices, allowing for the representation of fractional statistics and enhanced symmetry protection. This overcomes limitations of purely Majorana-based codes, which often struggle to capture the full complexity of fermionic interactions and symmetry constraints, and establishes a unified description encompassing all Abelian fermionic topological orders possessing gapped boundaries and all supercohomology fermionic symmetry-protected topological phases. Supercohomology groups provide a mathematical framework for classifying these symmetry-protected phases, ensuring a rigorous and complete description. Scientists at Peking University and Stony Brook University have successfully demonstrated control over anyons, string operators, and braiding statistics through stabilizer algebra. The ability to manipulate these topological excitations is essential for implementing quantum gates and performing fault-tolerant quantum computation. Fermionic versions of clock and shift operators were introduced, providing a means to move and exchange anyons within the system while preserving the topological order. This allowed for an exact bosonization map for mathbb Z’DF symmetries when is even, meaning that the fermionic symmetry can be effectively mapped onto a bosonic symmetry under certain conditions. This map simplifies the analysis and allows for the application of well-established bosonic techniques. Furthermore, the team realised a nontrivial mathbb Z8F fermionic SPT phase lacking a free-fermion analogue, demonstrating the power of this new approach to generate phases of matter that are inaccessible through conventional methods. This provides a concrete example of the power of this new approach and highlights its potential for discovering novel quantum phases. Majorana-Pauli codes unlock modelling of fermionic topological phases Constructing exact models of intrinsically fermionic topological phases, materials exhibiting exotic quantum properties such as fractionalized excitations and protected edge states, has lagged behind progress with their bosonic counterparts. The difficulty arises from the inherent complexity of dealing with fermionic statistics and the lack of suitable mathematical tools for describing these systems. Scientists at Peking University and Stony Brook University have bridged this gap with Majorana-Pauli stabilizer codes, a technique combining standard Pauli operators, which describe the behaviour of electrons, and ghostly Majorana operators to create solvable models on a lattice. This advancement bridges a vital gap between theoretical understanding and practical modelling, potentially accelerating progress in both quantum materials science and quantum computing.

The team established a new framework for precisely modelling intrinsically fermionic topological phases, materials exhibiting unusual quantum properties, systematically combining conventional and ghostly Majorana operators within a lattice structure. This overcomes longstanding challenges in describing these complex systems algebraically and provides a unified description encompassing all Abelian fermionic topological orders with stable edges and all supercohomology fermionic symmetry-protected topological phases, offering a flexible platform for future research. The ability to accurately model these phases is crucial for predicting their properties and designing materials that exhibit them. The researchers successfully demonstrated a new method for modelling intrinsically fermionic topological phases using Majorana-Pauli stabilizer codes. This is significant because constructing exact models of these materials has proven more difficult than for their bosonic counterparts due to the complexity of fermionic behaviour. By combining Pauli and Majorana operators, the team created a solvable framework that accurately describes these phases and encompasses all Abelian fermionic topological orders with stable edges. The authors further extended this construction to include a specific example, a mathbb Z8F fermionic symmetry-protected topological phase without a free-fermion analogue. 👉 More information🗞 Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases✍️ Meng Sun, Zongyuan Wang, Nathanan Tantivasadakarn and Yu-An Chen🧠 ArXiv: https://arxiv.org/abs/2606.25048 Stay current. See today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals. Tags: