Meng Sun of Peking University and colleagues have introduced Majorana-Pauli stabilizer codes, a new class of exactly solvable fermionic lattice models constructed from generalized Pauli operators and Majorana operators. This work addresses a gap in the field, as systematic stabilizer descriptions of intrinsically fermionic topological phases have historically been less developed than those for their bosonic counterparts.
As a demonstration, the researchers constructed an exactly solvable stabilizer realization of the fermionic toric code, using Z8 Pauli operators coupled to Majorana modes. This approach extends the stabilizer-code paradigm, bridging fermionic quantum many-body physics to quantum error correction and connecting diverse topological phases within a common framework.
Majorana-Pauli Codes Define Fermionic Topological Phases
This realization extends beyond previous methods by directly implementing anyon condensation at the stabilizer level on a square lattice, offering a new level of control over the system’s behavior. The specific combination of Z8 operators and Majorana modes allows for a simplified stabilizer representation, particularly for fermionic symmetry-protected topological phases that lack a free-fermion realization. These codes provide exact lattice realizations, not just approximations, of bosonic topological orders and now their fermionic equivalents.
The ability to construct these codes stems from combining Z8 Pauli operators with Majorana operators, a technique detailed in the work and enabling direct implementation of anyon condensation. This approach allows for the creation of subsystem codes potentially corresponding to chiral Abelian fermionic phases, opening avenues for exploring more complex topological phenomena. The fermionic toric code, central to this work, is shown to be part of a broader “duality web” connecting diverse topological phases.
This web encompasses bosonic topological orders, symmetry-enriched topological phases and both bosonic and fermionic symmetry-protected topological phases, all within a unified stabilizer framework. This interconnectedness suggests a deeper underlying structure governing these phases, potentially simplifying their classification and understanding. The researchers state, highlighting the core mechanism driving this unification.
Fermionic Toric Code Realized with Z_8 Pauli & Majorana Operators
This implementation moves beyond approximations by providing a lattice-based model directly built on the interplay between these specific operators and particles, a departure from previous methods in the field. The resulting model accurately represents the fermionic toric code, a Z2 topological order in (2+1) dimensions, within a stabilizer framework. This approach extends the capabilities of stabilizer codes, traditionally used for bosonic topological orders, to encompass intrinsically fermionic systems.
The researchers demonstrate this by building a lattice realization of the fermionic toric code, a system where the anyons, string operators, fusion rules, and braiding statistics all emerge naturally from the stabilizer algebra. This means the code’s behavior is predictable and verifiable through the mathematical properties of its stabilizers. The researchers illustrate this interconnectedness with diagrams showing how different phases can be transformed into one another through processes like anyon condensation and gauging or ungauging symmetries.
Stabilizer Framework Controls Full Spectrum of Fermionic Phases
Stabilizer models offer direct algebraic control over a system’s full spectrum by organizing eigenstates according to stabilizer eigenvalues and mapping them with Pauli operators, a feature lacking in many commuting-projector models that primarily describe ground-state sectors. This capability extends to intrinsically fermionic topological phases, systems requiring physical fermions, where explicit commuting-projector Hamiltonians exist but a concise stabilizer description has remained elusive until now. The development of Majorana-Pauli stabilizer codes provides this missing framework, offering a unified lattice structure for both fermionic Abelian phases and their bosonic counterparts.
These new codes establish a duality web connecting diverse topological phases through operations like anyon condensation and the gauging or ungauging of bosonic and fermionic symmetries. This interconnectedness is visually represented in diagrams outlining the phases constructed and the operations linking them, detailed in sections VI of the research.
The commutativity of this diagram encapsulates the central finding. The unified stabilizer framework realizes and relates these phases through explicit condensation, gauging and ungauging. Gauging a symmetry, for instance, is achieved by coupling a physical fermion to a Z2 gauge field. Four purely bosonic phases, the trivial Z8 SPT phase, the Z4 SPT phase, the Z8 toric code, and the U(1)4× U(1)-4 theory, already have Pauli-stabilizer realizations, as described in a prior reference. This framework allows for accessible analysis of anyon content, symmetry enrichment and duality relations within a stabilizer code formalism.
Duality Webs Connect Bosonic & Fermionic Topological Orders
Researchers demonstrate that the fermionic toric code, a key example of an intrinsically fermionic topological order, exists within a broader duality web linking diverse phases of matter. This construction extends beyond simple connections, providing a common stabilizer description for all these phases, simplifying their analysis and comparison. The researchers’ work yields stabilizer realizations of all two-dimensional Abelian fermionic topological orders possessing fully gapped boundaries. The approach uses a method of binding an emergent fermion to a physical fermion, then condensing the resulting boson to achieve these results.
This process offers a simpler realization than previous string-net constructions, with an explicit stabilizer structure that clarifies the underlying mechanisms. To accommodate more general fermionic symmetries, the team introduced fermionic clock and shift operators extending the Majorana algebra, enabling a stabilizer realization of the nontrivial SPT phase, a phase lacking a free-fermion realization. These models are connected to bosonic topological orders and other SPT and SET phases through the condensation and gauging operations summarized in the paper’s figures.
Condensation of fψ Creates Intrinsic Fermionic Order
The researchers detail how this bare condensation differs from ungauging, a related operation that also removes intrinsic topological order but preserves a dual 0-form symmetry. The construction extends to include fermionic clock and shift operators defined on local Hilbert spaces consisting of qudits and complex fermions, allowing for manipulation of even-order systems.
Ungauging Symmetries Generates SET & SPT Phases
This unification is achieved through a common stabilizer description, allowing researchers to map relationships between these phases using anyon condensation and the manipulation of symmetries via gauging and ungauging. These codes incorporate both generalized Pauli operators and Majorana operators, offering a new algebraic transparency for analyzing complex systems. Ungauging, a process of removing symmetry constraints, contributes to generating both SET and SPT phases from the fermionic toric code.
Z_D^F Symmetries and Fermionic Clock/Shift Operators
The construction of fermionic clock and shift operators extends the utility of Majorana-Pauli stabilizer codes beyond previously explored systems, enabling an exact bosonization map for symmetries when D is even. These newly defined operators, denoted Γ and Π, combine bosonic and fermionic degrees of freedom, establishing a framework for analyzing intrinsically fermionic topological orders with gapped boundaries.
The operators’ commutation relations, ΓΠ = ζ ΠΓ and Γ’Π = ζ-1 ΠΓ’, are explicitly defined and verified using the established definitions of Γ, Γ’ and Π. The algebra governing these operators simplifies to the standard Majorana algebra when D equals 1, demonstrating a fundamental connection between the generalized and established frameworks. A specific example illustrates this further; when D is 4, the resulting operators satisfy Π2 = P, Γ2 = Z and Γ’2 = Z†, ΓΠ = iΠΓ, Γ’Π = -iΠΓ’.
These relationships align with previously published work on fermionic clock algebras and related constructions using “spinful fermions”. Expanding the scope to D = 8, where the bosonic qudit is a Z4 qudit, the operators are defined as Γ = γ, Γ’ = † γ’ and Π =. This specific instantiation highlights the versatility of the approach, allowing for the construction of a stabilizer model for a nontrivial fermionic symmetry-protected topological phase lacking a free-fermion analog.
The operators’ properties, Πd = P, ΠD = 1, Γ2 = Z, Γ’2 = Z†, ΓD = Γ’D = 1, demonstrate a complete and consistent algebraic structure. The ability to define and manipulate these fermionic clock and shift operators provides a powerful tool for exploring fermionic topological phases.
Stabilizer Codes Bridge Quantum Error Correction & Fermionic Physics
The construction of Majorana-Pauli stabilizer codes extends the reach of stabilizer-code paradigms into intrinsically fermionic phases, offering a new approach to bridging fermionic quantum many-body physics and quantum error correction. This specific instantiation provides a concrete example of how the framework can be applied to model established phases with a new set of tools.
The ability to manipulate these codes extends beyond simply modeling existing phases. The researchers suggest potential applications in quantum simulation of fermionic phases of matter, as well as their use as building blocks for quantum error correction and quantum computation in systems with native physical fermions.
The work builds on the success of stabilizer models in bosonic systems, which have already established themselves as a cornerstone of quantum error correction and a powerful framework for understanding bosonic topological phases. The team’s work suggests a path toward a unified microscopic framework for understanding a broader range of quantum phases and their potential for technological applications.
👉 More information
🗞 Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases
✍️ Meng Sun, Zongyuan Wang, Nathanan Tantivasadakarn and Yu-An Chen
🧠 DOI: http://link.aps.org/doi/10.1103/kw13-xkkb
See today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals.
