This paper proposes a cryptographic framework whose core security foundationrests upon the physical process of anyon braiding in topological quantum computation. In two-dimensional topological materials, the fusion and braiding operationsof non-Abelian anyons are strictly protected by topological order: local perturbations cannot alter the fusion channels of anyon pairs, and braiding operationsdepend solely on the homotopy class of the braiding trajectory. This topologicalprotection mechanism provides a security resource for cryptography that is fundamentally different from computational complexity assumptions. The central ideaof this paper is to directly map anyon braiding operations into cryptographic primitives: the private key corresponds to the secret braiding sequences held by Aliceand Bob, respectively, and the public key consists of exchanged auxiliary anyonsand their partial observables. Security is rooted in the principle of topologicalprotection—any eavesdropper who has not executed the exact homotopy class ofbraiding operations cannot extract the information encoded in the fusion spacewithout destroying the ground-state degeneracy of the system. This paper constructs a non-interactive key exchange protocol based on anyon braiding, analyzesits inherent fault tolerance under physical noise, and discusses the prospects forimplementation on current physical platforms.
This paper proposes a cryptographic framework whose core security foundationrests upon the physical process of anyon braiding in topological quantum computation. In two-dimensional topological materials, the fusion and braiding operationsof non-Abelian anyons are strictly protected by topological order: local perturbations cannot alter the fusion channels of anyon pairs, and braiding operationsdepend solely on the homotopy class of the braiding trajectory. This topologicalprotection mechanism provides a security resource for cryptography that is fundamentally different from computational complexity assumptions. The central ideaof this paper is to directly map anyon braiding operations into cryptographic primitives: the private key corresponds to the secret braiding sequences held by Aliceand Bob, respectively, and the public key consists of exchanged auxiliary anyonsand their partial observables. Security is rooted in the principle of topologicalprotection—any eavesdropper who has not executed the exact homotopy class ofbraiding operations cannot extract the information encoded in the fusion spacewithout destroying the ground-state degeneracy of the system. This paper constructs a non-interactive key exchange protocol based on anyon braiding, analyzesits inherent fault tolerance under physical noise, and discusses the prospects forimplementation on current physical platforms.
This paper proposes a cryptographic framework whose core security foundation rests upon the topological rigidity of the modular data within modular fusioncategories. Modular fusion categories provide the complete mathematical rules forthe fusion and braiding of anyons in two-dimensional topological orders, and theirmodular data consist of the S-matrix and the T-matrix—the S-matrix describesthe mutual braiding statistics of anyons, and the T-matrix gives their topological spins. These two matrices together generate a projective representation of themodular group SL(2,Z) and are deeply coupled to the fusion rules via the Verlindeformula. The central idea of this paper is to transform this spectral data into acryptographic security resource: the private key corresponds to a simple object ina modular fusion category, and the public key is a partial truncation of the corresponding row of the object in the S-matrix. Security is rooted in the hardnessof modular data reconstruction—recovering the complete fusion rules and categorical structure from sparse row fragments of the S-matrix constitutes a highlyunderdetermined algebraic inverse problem, whose solution is equivalent, in theworst case, to solving large-scale nonlinear polynomial systems. This paper firstreveals the inherent circular dependency paradox in constructing a non-interactivekey exchange protocol directly based on S-matrix symmetry, and then proposes amodular data commitment framework. Through a commitment-challenge-responsetriple, the use of symmetry is transformed from “direct shared secret” to “bidirectional consistency verification”. On this basis, a two-round interactive key exchangeprotocol, MDC-KEX, is constructed, and a security reduction to the hardness ofmodular data reconstruction is provided in the random oracle model. This paperanalyzes the post-quantum resistance of modular data cryptography from threedimensions: algorithmics, algebraic structure, and physical implementation—thenon-commutative fusion ring of modular fusion categories renders Shor-type algorithms inapplicable, the NP-hard solving complexity of the pentagon equationsposes a fundamental barrier for quantum computers, and the physical measurementorigin of the S-matrix provides cross-disciplinary security depth. Finally, based onthe classification data of known low-rank modular fusion categories, the parameterspace and computational feasibility are evaluated.
Indistinguishability of particles is a fundamental principle of quantum mechanics. For all elementary and quasiparticles observed to date - including fermions, bosons, and Abelian anyons - this principle guarantees that the braiding of identical particles leaves the system unchanged. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well developed mathematical description of non-Abelian anyons and numerous theoretical proposals, the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. While efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasi-particles, superconducting quantum processors allow for directly manipulating the many-body wavefunction via unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons, we implement a generalized stabilizer code and unitary protocol to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of employing the anyons for quantum computation and utilize braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and - through the future inclusion of error correction to achieve topological protection - could open a path toward fault-tolerant quantum computing.
FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group Bn generators σi satisfy σiσi+1σi=σi+1σiσi+1 (Yang–Baxter equation); non-Abelian anyons carry unitary representations U(σi) with U(σi)U(σj)=U(σj)U(σi) for ∣i−j∣≥2. Fibonacci anyons: fusion rule τ×τ=1+τ, Hilbert space dimension grows as Fibonacci sequence Fn, golden ratio ϕ=(1+5)/2≈1.618 appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-g surfaces: dimension =kg for level-k Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio Fn+1/Fn→1.618 — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
Topological quantum computation aims to encode quantum information in the nonlocal degrees of freedom of non-Abelian anyons and manipulate it through braiding and fusion operations， offering a promising pathway to fault-tolerant quantum computation. Non-Abelian anyons， with their nontrivial braiding statistics and intrinsic topological fault tolerance， are regarded as a key building block for realizing scalable and decoherence-resistant quantum computing. This review systematically surveys the key developments in anyon physics over the past four decades， from the early theoretical prediction of fractional statistics to direct experimental verification of anyon braiding in fractional quantum Hall states. The physical principles and key findings of essential experimental schemes， including Fabry-Pérot interferometers， Mach-Zehnder interferometers， and anyon colliders， are comprehensively covered. Based on these foundations， the characterization of topological order in even-denominator fractional quantum Hall states， exemplified by the ν=5/2 state， and transport and interferometric probes of non-Abelian statistics are discussed in detail. Furthermore， recent advances in topological quantum states in novel material platforms， including graphene and twisted transition metal dichalcogenides， are reviewed， together with artificial quantum platforms that simulate non-Abelian braiding using superconducting quantum processors. Finally， the key challenges and future directions on the path from anyon physics to scalable topological quantum computing prototype devices are outlined.
This paper proposes a cryptographic framework whose core security foundationrests upon the structural rigidity of fusion categories. Fusion categories are thecategorical generalization of representation categories of finite groups within theframework of tensor categories, and their classification and representation theoryare strictly constrained by Ocneanu’s rigidity theorem: given the fusion rules, themonoidal structure of the category admits only finitely many possibilities up toequivalence, and is completely determined by the coherence equations of the Fsymbols. This rigidity makes the modular data of a fusion category—the S-matrixand T-matrix—ideal cryptographic invariants. This paper formally defines two corehardness assumptions: Modular Data Reconstruction Hardness and Fusion RuleDecomposition Hardness. Based on these, two types of trapdoor constructions aredesigned to resolve the public-private computational asymmetry: a lookup-tablemethod relying on a complete categorical database, and a vector recovery methodbased on one-way projection from a standard form. A non-interactive key exchangeprotocol is constructed, and a security reduction is completed in the random oraclemodel. Furthermore, the post-quantum resistance of the framework is systematically analyzed from three perspectives: algorithmic, topological, and physical,with the square-root speedup of Grover’s quantum search counteracted by a multicategory direct product composition strategy. Finally, based on the classificationdata of known modular fusion categories, the parameter space and computationalfeasibility are evaluated, providing a self-consistent theoretical foundation for fusioncategory cryptography to move from theoretical conception to engineering practice