non-Abelian Braiding Research Articles

Braids and higher-order exceptional points from the interplay between lossy defects and topological boundary states

We show that the perturbation of the Su-Schrieffer-Heeger chain by a localized lossy defect leads to higher-order exceptional points (HOEP). Depending on the location of the defect, third- and fourth-order exceptional points (EP3 and EP4) appear in the space of Hamiltonian parameters. On the one hand, they arise due to the non-Abelian braiding properties of exceptional lines (EL) in parameter space. Namely, the HOEPs lie at intersections of mutually noncommuting ELs. On the other hand, we show that such special intersections happen due to the fact that the delocalization of edge states, induced by the non-Hermitian defect, hybridizes them with defect states. These can then coalesce together into an EP3. When the defect lies at the midpoint of the chain, a special symmetry of the full spectrum can lead to an EP4. In this way, our model illustrates the emergence of interesting non-Abelian topological properties in the multiband structure of non-Hermitian perturbations of topological phases. Published by the American Physical Society 2024

Non-Abelian braiding in three-fold degenerate subspace and the acceleration

Non-Abelian braiding operations of quantum states have attracted substantial attention due to their great potentials for realizing topological quantum computations. The adiabatic version of quantum braiding is robust against systematic errors, yet will suffer from decoherence and dephasing effects due to a long evolution time. In this paper, we propose to realize the braiding process in a three-fold degenerate subspace of a seven-level system, where the non-Abelian effect can be detected by changing the orders of the braiding. We accelerate the adiabatic control through adding auxiliary coupling terms according to a shortcut to adiabatic theory for the non-Abelian case. Furthermore, by generalizing the parallel adiabatic passages, adiabatic control can be accelerated through only reshaping the original control waveforms and the effective pulses area will be significantly reduced. Therefore, the proposed schemes may provide an experimentally feasible way to investigate the non-Abelian braiding in atomic systems and the waveguide systems.

Progress in Superconductor-Semiconductor Topological Josephson Junctions

Majorana bound states (MBSs) are quasiparticles that are their own antiparticles. They are predicted to emerge as zero-energy modes localized at the boundary of a topological superconductor. No intrinsic topological superconductor is known to date. However, by interfacing conventional superconductors and semiconductors with strong spin-orbit coupling, it is possible to create a system hosting topological states. Hence, epitaxial superconductors and semiconductors have emerged as an attractive material system with atomically sharp interfaces and broad flexibility in device fabrications incorporating Josephson junctions. We discuss the basics of topological superconductivity and provide insight into how to go beyond current state-of-the-art experiments. We argue that the ultimate success in realizing MBS physics requires the observation of non-Abelian braiding and fusion experiments. Published by the American Physical Society 2024

The centre of a finitely generated strongly verbally closed group is almost always pure

ABSTRACT The assertion in the title implies that many interesting groups (for example, all non-abelian braid groups or $\textbf{SL}_{100}(\mathbb{Z})$) are not strongly verbally closed, i. e., they embed into some finitely generated groups as verbally closed subgroups, which are not retracts.

Non-Abelian braiding of Fibonacci anyons with a superconducting processor

Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.

Non-Abelian Braiding of Topological Edge Bands.

Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding structures, as demonstrated recently in a type of topological insulators known as Möbius insulators, whose topological edge states form two braided bands exhibiting a Möbius twist. While the formation of Möbius twist is inspiring, it belongs to the simple Abelian braid group B_{2}. The most fascinating features about topological braids rely on the non-Abelianness in the higher-order braid group B_{N} (N≥3), which necessitates multiple edge bands, but so far it has not been discussed. Here, based on the gauge enriched symmetry, we develop a scheme to realize non-Abelian braiding of multiple topological edge bands. We propose tight-binding models of topological insulators that are able to generate topological edge states forming non-Abelian braiding structures. Experimental demonstrations are conducted in two acoustic crystals, which carry three and four braided acoustic edge bands, respectively. The observed braiding structure can correspond to the topological winding in the complex eigenvalue space of projective translation operator, akin to the previously established point-gap winding topology in the bulk of the Hatano-Nelson model. Our Letter also constitutes the realization of non-Abelian braiding topology on an actual crystal platform, but not based on the "virtual" synthetic dimensions.

Generalized Kitaev spin liquid model and emergent twist defect

The Kitaev spin liquid model on a honeycomb lattice offers an intriguing feature that encapsulates both Abelian and non-Abelian phases (Alexei Kitaev, 2006). Recent studies suggest that the comprehensive phase diagram of the possible generalized Kitaev model largely depends on the specific details of the discrete lattice, which somewhat deviates from the traditional understanding of “topological” phases. In this paper, we propose an adapted version of the Kitaev spin liquid model on arbitrary planar lattices. Our revised model recovers the surface code model under certain parameter selections within the Hamiltonian terms. Changes in parameters can initiate the emergence of holes, domain walls, or twist defects. Notably, the twist defect, which presents as a lattice dislocation defect, exhibits non-Abelian braiding statistics upon tuning the coefficients of the Hamiltonian. Additionally, we illustrate that the creation, movement, and fusion of these defects can be accomplished through natural time evolution by linearly interpolating the static Hamiltonian. These defects demonstrate the Ising anyon fusion rule as anticipated. Our findings hint at possible implementation in actual physical materials owing to a more realistically achievable two-body interaction.

Three-dimensional PT -symmetric topological phases with a Pontryagin index

We report on a certain class of three-dimensional topological insulators and semimetals protected by spinless PT symmetry, hosting an integer-valued bulk invariant. We show using homotopy arguments that these phases host multigap topology, providing a realization of a single Z invariant in three spatial dimensions that is distinct from the Hopf index. We identify this invariant with the Pontryagin index, which describes Belavin-Polyakov-Schwartz-Tyupkin (BPST) instantons in particle physics contexts and corresponds to a three-sphere winding number. We study naturally arising multigap linked nodal rings, topologically characterized by split-biquaternion charges, which can be removed by non-Abelian braiding of nodal rings, even without closing a gap. We additionally recast the describing winding number in terms of gauge-invariant combinations of non-Abelian Berry connection elements, indicating relations to Pontryagin characteristic class in four dimensions. These topological configurations are furthermore related to fully nondegenerate multigap phases that are characterized by a pair of winding numbers relating to two isoclinic rotations in the case of four bands and can be generalized to an arbitrary number of bands. From a physical perspective, we also analyze the edge states corresponding to this Pontryagin index as well as their dissolution subject to the gap-closing disorder. Finally, we elaborate on the realization of these novel non-Abelian phases, their edge states, and linked nodal structures in acoustic metamaterials and trapped-ion experiments. Published by the American Physical Society 2024

Non-Abelian physics in light and sound.

Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.

Non-Abelian topological order and anyons on a trapped-ion processor.

Non-Abelian topological order is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged1-4. These anyonic excitations are promising building blocks of fault-tolerant quantum computers5,6. However, despite extensive efforts, non-Abelian topological order and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian topological order. Here we present the realization of non-Abelian topological order in the wavefunction prepared in a quantum processor and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum's H2 trapped-ion quantum processor, we create the ground-state wavefunction of D4 topological order on a kagome lattice of 27 qubits, with fidelity per site exceeding 98.4 per cent. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunnelling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon-a peculiar feature of non-Abelian topological order. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices.

Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems

While a significant fraction of topological materials has been characterized using symmetry requirements1–4, the past two years have witnessed the rise of novel multi-gap dependent topological states5–9, the properties of which go beyond these approaches and are yet to be fully explored. Although already of active interest at equilibrium10–15, we show that the combination of out-of-equilibrium processes and multi-gap topological insights galvanize a new direction within topological phases of matter. We show that periodic driving can induce anomalous multi-gap topological properties that have no static counterpart. In particular, we identify Floquet-induced non-Abelian braiding, which in turn leads to a phase characterized by an anomalous Euler class, being the prime example of a multi-gap topological invariant. Most strikingly, we also retrieve the first example of an ‘anomalous Dirac string phase’. This gapped out-of-equilibrium phase features an unconventional Dirac string configuration that physically manifests itself via anomalous edge states on the boundary. Our results not only provide a stepping stone for the exploration of intrinsically dynamical and experimentally viable multi-gap topological phases, but also demonstrate periodic driving as a powerful way to observe these non-Abelian braiding processes notably in quantum simulators.

The suppression of finite size effect within a few lattice sites

Boundary modes localized on the boundaries of a finite-size lattice experience a finite size effect (FSE) that could result in unwanted couplings, crosstalks and formation of gaps even in topological boundary modes. It is commonly believed that the FSE decays exponentially with the size of the system and thus requires many lattice sites before eventually becoming negligibly small. Here we consider a two-dimensional strip geometry that is periodic along one direction and truncated along the other direction, in which we identify a special type of FSE of some boundary modes that apparently vanishes at some particular wave vectors along the periodic direction. Meanwhile, the number of wave vectors where the FSE vanishes equals the number of lattice sites across the strip. We analytically prove this type of FSE in a simple model and prove this peculiar feature. We also provide a physical system consisting of a plasmonic sphere array where this FSE is present. Our work points to the possibility of almost arbitrarily tunning of the FSE, which facilitates unprecedented manipulation of the coupling strength between modes or channels such as the integration of multiple waveguides and photonic non-abelian braiding.

Electronic Structures of Kitaev Magnet Candidates RuCl3 and RuI3.

Layered honeycomb magnets with strong atomic spin-orbit coupling at transition metal sites have been intensively studied for the search of Kitaev magnetism and the resulting non-Abelian braiding statistics. α-RuCl3 has been the most promising candidate, and there have been several reports on the realization of sibling compounds α-RuBr3 and α-RuI3 with the same crystal structure. Here, we investigate correlated electronic structures of α-RuCl3 and α-RuI3 by employing first-principles dynamical mean-field theory. Our result provides a valuable insight into the discrepancy between experimental and theoretical reports on transport properties of α-RuI3, and suggests a potential realization of correlated flat bands with strong spin-orbit coupling and a quantum spin-Hall insulating phase in α-RuI3.

Gain-loss-induced non-Abelian Bloch braids

Onsite gain-loss-induced topological braiding principle of non-Hermitian energy bands is theoretically formulated in multiband lattice models with Hermitian hopping amplitudes. Braid phase transition occurs when the gain-loss parameter is tuned across exceptional point degeneracy. Laboratory realizable effective-Hamiltonians are proposed to realize braid groups B2 and B3 of two and three bands, respectively. While B2 is trivially Abelian, the group B3 features non-Abelian braiding and energy permutation originating from the collective behavior of multiple exceptional points. Phase diagrams with respect to lattice parameters to realize braid group generators and their non-commutativity are shown. The proposed theory is conducive to synthesizing exceptional materials for applications in topological computation and information processing.

A Hierarchy in Majorana Non-Abelian Tests and Hidden Variable Models

The recent progress of the Majorana experiments paves a way for the future tests of non-Abelian braiding statistics and topologically protected quantum information processing. However, a deficient design in those tests could be very dangerous and reach false-positive conclusions. A careful theoretical analysis is necessary so as to develop loophole-free tests. We introduce a series of classical hidden variable models to capture certain key properties of Majorana system: non-locality, topologically non-triviality, and quantum interference. Those models could help us to classify the Majorana properties and to set up the boundaries and limitations of Majorana non-Abelian tests: fusion tests, braiding tests and test set with joint measurements. We find a hierarchy among those Majorana tests with increasing experimental complexity.

Non-Abelian anyons with Rydberg atoms

We study the emergence of topological matter in two-dimensional systems of neutral Rydberg atoms in Ruby lattices. While Abelian anyons have been predicted in such systems, non-Abelian anyons, which would form a substrate for fault-tolerant quantum computing, have not been generated. To generate anyons with non-Abelian braiding statistics, we consider systems with mixed-boundary punctures. We obtain the topologically distinct ground states of the system numerically using the infinite Density Matrix Renormalization Group technique. We discuss how these topological states can be created using ancilla atoms of a different type. We show that a system with $2N+2$ punctures and an equal number of ancilla atoms leads to $N$ logical qubits whose Hilbert space is determined by a set of stabilizing conditions on the ancilla atoms. Quantum gates can be implemented using a set of gates acting on the ancilla atoms that commute with the stabilizers and realize the braiding group of non-Abelian Ising anyons.

Non-Abelian braiding of graph vertices in a superconducting processor

Indistinguishability of particles is a fundamental principle of quantum mechanics1. 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 unchanged2,3. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions4–8. 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 proposals9–22, 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. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons9,10, we implement a generalized stabilizer code and unitary protocol23 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 using the anyons for quantum computation and use 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 towards fault-tolerant quantum computing.

Digital Simulation of Projective Non-Abelian Anyons with 68 Superconducting Qubits

Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons, i.e., the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.

Composite Fermion Pairing Induced by Landau Level Mixing.

Pairing of composite fermions provides a possible mechanism for fractional quantum Hall effect at even denominator fractions and is believed to serve as a platform for realizing quasiparticles with non-Abelian braiding statistics. We present results from fixed-phase diffusion MonteCarlo calculations which predict that substantial Landau level mixing can induce a pairing of composite fermions at filling factors ν=1/2 and ν=1/4 in the l=-3 relative angular momentum channel, thereby destabilizing the composite-fermion Fermi seas to produce non-Abelian fractional quantum Hall states.

Recent progress on non-Abelian anyons: from Majorana zero modes to topological Dirac fermionic modes

Majorana zero modes (MZMs) have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation (TQC). In addition to the Majorana scheme, some non-Majorana quasiparticles obeying non-Abelian statistics, including topological Dirac fermionic modes, have also been proposed and therefore become new candidates for TQC. In this review, we overview the non-Abelian braiding properties as well as the corresponding braiding schemes for both the MZMs and the topological Dirac fermionic modes, emphasizing the recent progress on topological Dirac fermionic modes. A topological Dirac fermionic mode can be regarded as a pair of MZMs related by unitary symmetry, which can be realized in a number of platforms, including the one-dimensional topological insulator, higher-order topological insulator, and spin superconductor. This topological Dirac fermionic mode possesses several advantages compared with its Majorana cousin, such as superconductivity-free and larger gaps. Therefore, it provides a new avenue for investigating non-Abelian physics and possible TQC.