Anyons reveal a regime of quantum mechanics in which the history of how particles move around one another becomes a physical observable, turning topology into a new form of particle statistics and a possible architecture for quantum computation.
Every elementary particle encountered in ordinary three-dimensional physics belongs to one of two statistical families. Fermions, including electrons, protons, and quarks, obey the Pauli exclusion principle and cannot freely occupy the same quantum state. Bosons, including photons and many collective excitations, behave differently and can accumulate into the same state. This division is so fundamental that it is built into quantum field theory through the distinction between integer and half-integer spin. Yet the familiar boson-fermion dichotomy contains a hidden assumption about space itself. When quantum particles are restricted to move in two spatial dimensions, the topology of their possible trajectories changes, and quantum mechanics allows a much broader class of particle statistics. The resulting excitations are called anyons.
Anyons are not hypothetical replacements for electrons or photons moving freely through the vacuum. They appear as collective quasiparticles in special two-dimensional quantum phases, most famously fractional quantum Hall states. What distinguishes them is not merely an unusual charge or energy. Their quantum state can depend on how one anyon has moved around another. In the simplest case, winding one anyon around another changes the phase of the many-body wave function by an amount that is neither the bosonic nor fermionic value. For more exotic non-Abelian anyons, braiding particles can transform the system into an entirely different state within a degenerate quantum space. The topology of the trajectories becomes part of the physics.
This is one of the clearest examples of the principle introduced by topological quantum matter: quantum mechanics can make global mathematical structure experimentally observable. With anyons, however, the relevant structure does not live primarily in an electronic band or momentum-space invariant. It is encoded in the paths particles trace through space and time.
The reason two dimensions are special can be understood by comparing particle exchanges. Imagine two identical particles in three-dimensional space. If one particle is exchanged with the other, their trajectories can be continuously rearranged without encountering an obstruction. Performing the exchange twice produces a path that can ultimately be untangled back into a trivial motion. Quantum mechanics therefore permits the wave function to return either unchanged or with a minus sign after exchange, corresponding to bosons and fermions.
In two dimensions, the situation changes because particles cannot simply move above or below one another. One particle can wind clockwise around another or counterclockwise around it, and those paths cannot necessarily be smoothly deformed into one another without forcing the particles to pass through the same point. Successive exchanges therefore form distinct topological objects. Mathematically, particle permutations are replaced by elements of the braid group, which retains information about how trajectories wind around each other rather than recording only the final positions of the particles.
This distinction is subtle but fundamental. Ordinary permutation statistics care mainly about where identical particles end up. Anyonic statistics can care about the topology of the process that brought them there. Two experiments can begin with the same quasiparticles and finish with them occupying the same locations, yet produce different quantum states because the worldlines between those endpoints form different braids. Topology has transformed particle history into a physical degree of freedom.
The first physically important setting for anyons emerged from the fractional quantum Hall effect. When electrons are confined to a two-dimensional layer, cooled to very low temperatures, and exposed to a strong perpendicular magnetic field, interactions can reorganize the electrons into collective quantum liquids with fractionally quantized Hall conductance. Robert Laughlin's theory of the one-third fractional quantum Hall state showed that the elementary excitations of this collective state carry a fraction of the electron's charge rather than the charge of a conventional electron. Subsequent theoretical work revealed that their statistics are fractional as well. In 1984, Daniel Arovas, John Robert Schrieffer, and Frank Wilczek calculated the quantum phase accumulated when one fractional quantum Hall quasiparticle is transported around another and showed that it obeys anyonic statistics.
This provides an important lesson about quasiparticles. An anyon does not need to correspond to one fundamental constituent of matter. It can emerge from correlations among an enormous number of electrons. The underlying electrons remain fermions, but the collective excitations of the entire strongly correlated state obey a different effective statistics.
The phenomenon is analogous to other forms of emergence in condensed-matter physics, but considerably more radical. A phonon behaves as a quantum of collective lattice vibration even though no individual atom inside the crystal is a phonon. An anyon similarly represents an excitation of a many-electron quantum state, except that its effective charge and exchange statistics can possess properties unavailable to the microscopic electrons themselves. The many-body system has created a new kind of particle.
Experiments face a difficult problem because particle statistics are not usually measured by watching trajectories directly. Anyons inside fractional quantum Hall materials are microscopic collective excitations, and their defining property is a phase in the quantum wave function. Researchers therefore use interferometry to translate that invisible phase into a measurable electrical signal.
The idea resembles optical interference. A quantum quasiparticle can travel along alternative paths around an interferometer before the alternatives recombine. The measured interference depends on the relative quantum phase accumulated along those paths. If one path winds around localized anyons while another does not, the statistical phase generated by the braid can shift the interference pattern.
In 2020, an electronic Fabry-Pérot interferometer operating in the one-third fractional quantum Hall state provided direct experimental evidence of this effect. Researchers observed discrete phase shifts consistent with the expected anyonic braiding phase when quasiparticles entered or left the region enclosed by the interfering edge trajectory.
The experiment illustrates how topology becomes an electrical measurement. An abstract distinction between two worldline configurations changes the phase of a many-body wave function, which changes an interference pattern, which ultimately changes the current measured through a device.
Experimental control has advanced considerably since those early interferometric demonstrations. In 2025, graphene-based fractional quantum Hall interferometers were used to study braiding through discrete changes in interference associated with individual quasiparticles entering and leaving the interferometer. Other experiments that year demonstrated anyonic braiding in a chiral Mach-Zehnder geometry and observed coherent bunching and dissociation phenomena that reflect the collective statistics of fractional quasiparticles.
A 2026 experiment pushed the same interferometric program further by examining the slow dynamics of localized quasiparticles inside a fractional quantum Hall Fabry-Pérot interferometer. The device allowed researchers to distinguish changes associated with the anyonic statistical phase while simultaneously exposing how slowly the localized quasiparticle configuration can equilibrate. The result demonstrates both the progress and difficulty of anyon experiments: the topological phase may be sharply defined, but extracting it requires understanding electrostatics, interference, charge relaxation, and the detailed thermodynamics of the device.
Modern anyon experiments are therefore moving beyond simply asking whether fractional statistics exist. They are beginning to manipulate the quasiparticle number, control interference trajectories, study their dynamics, and investigate how collections of anyons behave together.
The particles observed in the simplest fractional quantum Hall states are Abelian anyons. Their braiding changes the wave function by a phase. The phase can be fractional and therefore fundamentally different from bosonic or fermionic exchange, but successive braids still combine in a way that does not depend on their ordering.
Non-Abelian anyons are more unusual. A collection of these quasiparticles can possess multiple quantum states that are locally indistinguishable from one another. Braiding the particles acts on this multidimensional state space. Instead of merely multiplying the wave function by a phase, a braid performs a transformation that can rotate one quantum state into a superposition of others. Crucially, different braids can perform transformations whose order matters. Braiding particles A and B followed by B and C can produce a different result from performing those operations in the opposite order. This noncommutativity is the origin of the term non-Abelian.
The information is stored nonlocally. It is not assigned to the internal state of one anyon in the way information might be stored in the spin of an electron. Instead, it belongs to the collective fusion space of several anyons. Bringing two anyons together can produce different possible total topological charges, and the allowed outcomes are described by fusion rules. Braiding changes amplitudes within this fusion space while preserving the relevant global topological information.
This changes the meaning of a quantum state. Two collections of quasiparticles can look locally identical at every particle position yet represent different quantum information because of how their collective topological charges are organized.
Among the most celebrated theoretical examples are Fibonacci anyons. Their name arises from the structure generated by repeatedly combining their fusion possibilities: the dimension of the relevant many-anyon state spaces grows according to Fibonacci-like counting. More importantly for quantum computation, the set of transformations generated by braiding Fibonacci anyons is rich enough to approximate arbitrary quantum gates. This makes Fibonacci topological order a canonical model for universal topological quantum computation.
The underlying mathematics connects quantum physics to braid groups, fusion categories, knot invariants, and topological quantum field theory. A braid describes the history of the anyons. Fusion rules specify how quasiparticles can combine. Associativity transformations relate different ways of grouping those fusions, while braiding transformations determine how exchanging particles acts on the fusion space. Consistency between these operations imposes powerful algebraic constraints.
This is why the common phrase that topological quantum computers “compute with braids” is more than visual imagery. The logical operation is represented mathematically by a braid-group action on a protected quantum-state space. Different microscopic trajectories that can be smoothly deformed into the same braid are intended to implement the same topological transformation.
This observation led Alexei Kitaev and others to propose topological quantum computation. Instead of encoding a qubit in one microscopic object whose state can be perturbed locally, quantum information can be distributed across the collective state of separated anyons. Computation is then performed through controlled braiding and fusion. Because the result depends ideally on the braid topology rather than every small geometric feature of the trajectory, some local control errors do not directly alter the logical operation.
The protection is not magical. Real topological systems have finite gaps, thermal excitations, imperfect measurements, quasiparticle poisoning, uncontrolled creation of excitations, and errors that can themselves form topologically nontrivial paths. Topological protection suppresses particular local error mechanisms; it does not eliminate the need for fault-tolerant architecture. The practical question is whether the physical error rates can be reduced enough that the topological encoding provides a decisive advantage over more conventional qubits. [3]
The conceptual advantage remains profound. In an ordinary control system, a gate can depend sensitively on the precise pulse amplitude, timing, or path through parameter space. In an ideal topological gate, a large family of microscopically different trajectories corresponds to the same braid. Geometry can fluctuate while topology remains fixed. This is quantum computation built around equivalence classes of physical histories.
There is an important distinction between finding non-Abelian anyons in a naturally occurring material and simulating their mathematics on a programmable quantum computer. Recent experiments have made remarkable progress on the second problem. In 2024, a trapped-ion processor was used to realize a non-Abelian topologically ordered state and manipulate its anyonic excitations, providing a controlled quantum simulation of non-Abelian fusion and braiding structure.
That same year, researchers using superconducting qubits realized the topological order of a Fibonacci string-net model and demonstrated non-Abelian braiding operations associated with Fibonacci anyons. The experiment created pairs of simulated Fibonacci anyons, applied braiding sequences, and measured the resulting fusion behavior.
In 2025, a related superconducting-processor experiment developed a scalable dynamical preparation of Fibonacci string-net states and demonstrated creation, identification, and braiding of Fibonacci anyons. The experiment also exploited a remarkable mathematical connection between Fibonacci string nets and chromatic polynomials, linking topological quantum matter to a difficult problem in graph theory.
These demonstrations are scientifically important because they allow researchers to experimentally manipulate topological orders whose natural material realization remains extremely challenging. They should not, however, be confused with discovering free-standing physical Fibonacci anyons in a condensed-matter sample. The processor's ordinary qubits are deliberately programmed to reproduce the many-body state and algebra of the desired topological theory. The experiment demonstrates the topological model and its operations, not yet a passive material whose native low-energy quasiparticles automatically supply the same hardware-level protection.
Anyons also reveal why topology is deeply connected to quantum field theory. At long wavelengths, many topologically ordered phases can be described by topological quantum field theories, in which the important observables depend less on local metric geometry than on global structures such as linking and braiding. Edward Witten famously connected three-dimensional Chern-Simons quantum field theory to the Jones polynomial of knots, demonstrating a profound relationship among quantum field theory, knot theory, and topology. Fibonacci-type anyon theories inherit this broader mathematical world, where particle worldlines and topological invariants are different representations of the same underlying structure.
This does not mean that an anyon is literally a knot. A braid consists of several strands whose endpoints remain distinct, while a knot is typically formed from a closed loop. The relationship emerges because worldlines in spacetime can form braided and linked structures, and closing those braids can generate knots and links whose invariants encode information about the corresponding topological theory.
The physical consequences can therefore be understood at several levels simultaneously. In condensed-matter language, one speaks about quasiparticles inside a strongly correlated electron system. In quantum-information language, one speaks about fusion spaces and protected logical states. In mathematics, one studies braid-group representations, fusion categories, and knot invariants. In quantum field theory, the same structures appear through topological field theories. Anyons sit precisely where these languages meet.
The current experimental frontier makes this connection more than a theoretical curiosity. Interferometers are increasingly able to probe Abelian anyonic phases directly in fractional quantum Hall systems, including increasingly sophisticated graphene and semiconductor devices. Programmable quantum processors can construct artificial topological states and execute non-Abelian braiding protocols that are far harder to isolate in natural materials. These are complementary approaches rather than interchangeable achievements. One attempts to reveal anyons as emergent excitations of matter; the other uses controlled quantum hardware to build and study their mathematical world.
The long-term challenge for non-Abelian topological quantum computation is to bring those two strengths together. Researchers would like a physical platform in which non-Abelian excitations arise robustly, can be created and separated, can be braided or otherwise manipulated with high fidelity, can be fused and measured reliably, and retain the topological protection that motivates the entire architecture. Theoretical candidates include special fractional quantum Hall states, topological superconducting systems, and quantum spin-liquid phases, while synthetic quantum simulators provide another route for studying the same underlying theories.
Whether this ultimately produces a scalable topological quantum computer remains unresolved. The physical significance of anyons does not depend on that outcome. Their existence has already altered one of the oldest classifications in quantum mechanics by showing that bosons and fermions are not the complete story once dimensionality and topology are taken seriously.
The deepest lesson of anyons is that a particle need not be characterized only by what it is at one instant. In two-dimensional quantum matter, the collective state can retain information about how particles have moved around one another. Two configurations with identical particle positions can represent different quantum states because their histories belong to different topological classes.
This is a fundamentally different way of storing physical information. Ordinary local properties can be changed by acting at a point. Topological information is distributed across relationships among trajectories and quasiparticles. To alter it, a process must change the relevant global structure.
That idea connects directly back to why topology has become so powerful throughout quantum physics. A Chern number records the global twisting of wave functions across momentum space. A topological edge state records the mismatch between two bulk phases. A Weyl node carries a topological charge in momentum space. Anyons extend the same philosophy into dynamics: their worldlines remember how particles have wound around one another.
In an anyonic system, that history can alter an interference experiment, transform a many-body quantum state, or implement a logical gate. The path itself becomes part of the information carried by matter, and particle motion becomes a form of topology made physical.