Topological Quantum Computation Research Articles

Dynamics of Fricke–Painlevé VI Surfaces

The symmetries of a Riemann surface Σ∖{ai} with n punctures ai are encoded in its fundamental group π1(Σ). Further structure may be described through representations (homomorphisms) of π1 over a Lie group G as globalized by the character variety C=Hom(π1,G)/G. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere Σ=S2(4) and the ‘space-time-spin’ group G=SL2(C). In such a situation, C possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface Va,b,c,d(x,y,z) with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or PVI); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of PVI. In this paper, we feature the parametric representation of some solutions of PVI: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups fp encountered in TQC or DNA/RNA sequences are proposed.

Majorana zero mode and its lattice construction in iron-based superconductors

Majorana zero modes (MZMs) obey non-Abelian braiding statistics. The braiding of MZMs can be used to construct the basic unit − topological qubit − of the topological quantum computation, which is immune to environmental noise and can achieve fault-tolerant quantum computation. The existing MZM platforms include hybrid structures such as topological insulator/superconductor, semiconducting nanowire/superconductor and 1d magnetic atomic chain/superconductor, and single materials such as 2M-WS<sub>2</sub>, 4Hb-TaS<sub>2</sub>, and iron-based superconductors (IBSs). The IBSs have advantages such as easy to fabricate, pure MZMs and high surviving temperatures of MZMs. Recently, a large-scale, ordered and tunable MZM lattice has been observed in LiFeAs, which provides a promising platform to future topological quantum computation. In this paper, first, we review the experimental observations of MZMs in IBSs, focusing on FeTe<sub>0.55</sub>Se<sub>0.45</sub>, (Li<sub>0.84</sub>Fe<sub>0.16</sub>)OHFeSe, CaKFe<sub>4</sub>As<sub>4</sub> and LiFeAs. Next, we introduce the critical experimental evidences of the MZMs. We also review the recent research work on the ordered and tunable MZM lattice in LiFeAs. Finally, we give conclusion and perspective on future Majorana research.

Fractional Quantum Hall State at Filling Factor ν=1/4 in Ultra-High-Quality GaAs Two-Dimensional Hole Systems.

Single-component fractional quantum Hall states (FQHSs) at even-denominator filling factors may host non-Abelian quasiparticles that are considered to be building blocks of topological quantum computers. Such states, however, are rarely observed in the lowest-energy Landau level, namely at filling factors ν<1. Here, we report evidence for an even-denominator FQHS at ν=1/4 in ultra-high-quality two-dimensional hole systems confined to modulation-doped GaAs quantum wells. We observe a deep minimum in the longitudinal resistance at ν=1/4, superimposed on a highly insulating background, suggesting a close competition between the ν=1/4 FQHS and the magnetic-field-induced, pinned Wigner solid states. Our experimental observations are consistent with the very recent theoretical calculations that predict that substantial Landau level mixing, caused by the large hole effective mass, can induce composite fermion pairing and lead to a non-Abelian FQHS at ν=1/4. Our results demonstrate that Landau level mixing can provide a very potent means for tuning the interaction between composite fermions and creating new non-Abelian FQHSs.

Next-generation even-denominator fractional quantum Hall states of interacting composite fermions.

The discovery of the fractional quantum Hall state (FQHS) in 1982 ushered a new era of research in many-body condensed matter physics. Among the numerous FQHSs, those observed at even-denominator Landau level filling factors are of particular interest as they may host quasiparticles obeying non-Abelian statistics and be of potential use in topological quantum computing. The even-denominator FQHSs, however, are scarce and have been observed predominantly in low-disorder two-dimensional (2D) systems when an excited electron Landau level is half filled. An example is the well-studied FQHS at filling factor [Formula: see text] 5/2 which is believed to be a Bardeen-Cooper-Schrieffer-type, paired state of flux-particle composite fermions (CFs). Here, we report the observation of even-denominator FQHSs at [Formula: see text] 3/10, 3/8, and 3/4 in the lowest Landau level of an ultrahigh-quality GaAs 2D hole system, evinced by deep minima in longitudinal resistance and developing quantized Hall plateaus. Quite remarkably, these states can be interpreted as even-denominator FQHSs of CFs, emerging from pairing of higher-order CFs when a CF Landau level, rather than an electron or a hole Landau level, is half-filled. Our results affirm enhanced interaction between CFs in a hole system with significant Landau level mixing and, more generally, the pairing of CFs as a valid mechanism for even-denominator FQHSs, and suggest the realization of FQHSs with non-Abelian anyons.

Topological magnetic line defects in Fe(Te, Se) high-temperature superconductors

The realization of Majorana zero modes in condensed matter have been attracting enormous interests from fundamental science such as topological quantum computation. Recently iron based superconductors were identified as a high-temperature platform for realizing topological superconductivity and Majorana modes. As unconventional superconductors, one of the most important characteristics of them is that they are in the vicinity of magnetic states due to the strong Hund’s coupling in iron atoms. Here we propose that the line defects with missing Te/Se anions in Fe(Se, Te) superconductors provide the realization of intrinsic antiferromagnetic (AFM) chains with Rashba spin-orbit coupling. Against conventional wisdom, Majorana zero modes (MZMs) can be robustly generated at these AFM chain ends. These results can consistently explain the recent experimental observation of zero-energy end states in line defects of monolayer Fe(Te, Se)/SrTiO3 by scanning tunneling microscopy (STM) measurements. Our research not only demonstrates an unprecedented interplay among native line defect, emergent magnetism and topological superconductivity but also explores a high-temperature platform for Majorana fermions.

Analysis of Majorana-based quantum computation

In recent years, the realization of quantum computation is a compelling research direction. To realize an error-tolerate quantum computer, topological quantum computation is considered as the most promising direction. Plenty of scholars have proved that non-abelian particles may play an important role on it. As a matter of fact, among particles obey non-abelian statistics, Majorana fermions is a typical example. On this basis, this study aims at introducing the development of Majorana-based quantum computation. To be specific, it will contain the conceptual framework of topological computation and introduce some recent research about Majorana-based quantum computation including the principles, configurations as well as the state-of-art analysis results. In addition, the current limitations for these studies will be demonstrated accordingly. Moreover, based on the summary the development of this field in recent years, this research will provide reference direction for future researchers. Overall, these results shed light on guiding further exploration of quantum computers.

The ZX-calculus as a language for topological quantum computation

Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.

Quantum computation at the edge of a disordered Kitaev honeycomb lattice

We analyze propagation of quantum information along chiral Majorana edge states in two-dimensional topological materials. The use of edge states may facilitate the braiding operation, an important ingredient in topological quantum computations. For the edge of the Kitaev honeycomb model in a topological phase, we discuss how the edge states can participate in quantum-information processing, and consider a two-qubit logic gate between distant external qubits coupled to the edge. Here we analyze the influence of disorder and noise on properties of the edge states and quantum-gate fidelity. We find that realistically weak disorder does not prevent one from implementation of a high-fidelity operation via the edge.

Experimental Simulation of Topological Quantum Computing with Classical Circuits

The key obstacle to the realization of a scalable quantum computer is overcoming environmental and control errors. Topological quantum computation attracts great attention because it emerges as one of the most promising approaches to solving these problems. Various theoretical schemes for building topological quantum computation have been proposed. However, experimental implementation has always been a great challenge because it has proved to be extremely difficult to create and manipulate topological qubits in real systems. Therefore, topological quantum computation has not been realized in experiments yet. Herein, the first experimental simulation of topological quantum computation with classical circuits is reported. Based on the proposed new scheme with circuits, not only Majorana‐like edge states are simulated experimentally, but also T junctions are constructed for simulating the braiding process. Furthermore, the feasibility of simulated topological quantum computing through a set of one‐ and two‐qubit unitary operations is demonstrated. Finally, the simulation of Grover's search algorithm demonstrates that simulated topological quantum computation is ideally suited for such tasks. The developed circuit‐based topological quantum‐computing simulator can provide important references for developing future topological quantum circuits.

Simulating the Hadamard gate in the Fibonacci disk code for universal topological quantum computation

Simulating the Hadamard gate in the Fibonacci disk code for universal topological quantum computation

Quantum double structure in cold atom superfluids

The theory of topological quantum computation is underpinned by two important classes of models. One is based on non-abelian Chern–Simons theory, which yields the so-called SU(2)k anyon models that often appear in the context of electrically charged quantum fluids. The physics of the other is captured by symmetry broken Yang–Mills theory in the absence of a Chern–Simons term and results in the so-called quantum double models. Extensive resources have been invested into the search for SU(2)k anyon quasi-particles, in particular, the so-called Ising anyons (k = 2) of which Majorana zero modes are believed to be an incarnation. In contrast to the SU(2)k models, quantum doubles have attracted little attention in experiments despite their pivotal role in the theory of error correction. Beyond topological error correcting codes, the appearance of quantum doubles has been limited to contexts primarily within mathematical physics, and as such, they are of seemingly little relevance for the study of experimentally tangible systems. However, recent works suggest that quantum double anyons may be found in spinor Bose–Einstein condensates. In light of this, the core purpose of this article is to provide a self-contained exposition of the quantum double structure, framed in the context of spinor condensates, by constructing explicitly the quantum doubles for various ground state symmetry groups and discuss their experimental realisability. We also derive analytically an equation for the quantum double Clebsch–Gordan coefficients from which the relevant braid matrices can be worked out. Finally, the existence of a particle-vortex duality is exposed and illuminated upon in this context.

Wiener–Hopf factorization approach to a bulk-boundary correspondence and stability conditions for topological zero-energy modes

Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight into the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener–Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland–Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener–Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener–Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero-energy modes are especially valuable for applications, including Majorana-based topological quantum computing.

Stencil Lithography‐Assisted Fabrication of Triangular Grid Pb Architecture

AbstractManipulating Majorana zero modes (MZMs) holds significant promise in propelling the field of topological quantum computation. While scanning tunneling microscopy (STM) exhibits remarkable capabilities in characterizing MZMs, additional geometrical constraints are imperative for achieving precise and controllable manipulation of these zero modes. To this end, it is of great help to integrate a micro mask within a molecular beam epitaxy (MBE) system to grow specific patterns containing MZMs. Here, a scenario for manipulating the MZMs in the vortex cores based on a triangular grid architecture is proposed. Additionally, a stencil lithography setup that is compatible with a combined MBE and STM system is introduced and experimentally demonstrated. A fine‐tuning of the Pb patterns, between a hexagonal antidot lattice to a triangular grid array, is achieved by adjusting the spatial gap between the stencil mask and the underlying substrate. This study provides new possibilities for the fabrication and investigation of Majorana devices.

Topological Superconductors from a Materials Perspective.

Topological superconductors (TSCs) have garnered significant research and industry attention in the past two decades. By hosting Majorana bound states which can be used as qubits that are robust against local perturbations, TSCs offer a promising platform toward (nonuniversal) topological quantum computation. However, there has been a scarcity of TSC candidates, and the experimental signatures that identify a TSC are often elusive. In this Perspective, after a short review of the TSC basics and theories, we provide an overview of the TSC materials candidates, including natural compounds and synthetic material systems. We further introduce various experimental techniques to probe TSCs, focusing on how a system is identified as a TSC candidate and why a conclusive answer is often challenging to draw. We conclude by calling for new experimental signatures and stronger computational support to accelerate the search for new TSC candidates.

Highly Anisotropic Even-Denominator Fractional Quantum Hall State in an Orbitally Coupled Half-Filled Landau Level.

The even-denominator fractional quantum Hall states (FQHSs) in half-filled Landau levels are generally believed to host non-Abelian quasiparticles and be of potential use in topological quantum computing. Of particular interest is the competition and interplay between the even-denominator FQHSs and other ground states, such as anisotropic phases and composite fermion Fermi seas. Here, we report the observation of an even-denominator fractional quantum Hall state with highly anisotropic in-plane transport coefficients at Landau level filling factor ν=3/2. We observe this state in an ultra-high-quality GaAs two-dimensional hole system when a large in-plane magnetic field is applied. By increasing the in-plane field, we observe a sharp transition from an isotropic composite fermion Fermi sea to an anisotropic even-denominator FQHS. Our data and calculations suggest that a unique feature of two-dimensional holes, namely the coupling between heavy-hole and light-hole states, combines different orbital components in the wave function of one Landau level, and leads to the emergence of a highly anisotropic even-denominator fractional quantum Hall state. Our results demonstrate that the GaAs two-dimensional hole system is a unique platform for the exploration of exotic, many-body ground states.

Gate-Defined Josephson weak-Links in Monolayer WTe2.

Systems combining superconductors with topological insulators offer a platform for the study of Majorana bound states and a possible route to realize fault tolerant topological quantum computation. Among the systems being considered in this field, monolayers of tungsten ditelluride (WTe2 ) have a rare combination of properties. Notably, it has been demonstrated to be a Quantum Spin Hall Insulator (QSHI) and can easily be gated into a superconducting state. Wereport measurements on gate-defined Josephson weak-link devices fabricated using monolayer WTe2 . It is found that consideration of the two dimensional superconducting leads are critical in the interpretation of magnetic interference in the resulting junctions. The reported fabrication procedures suggest a facile way to produce further devices from this technically challenging material and the results mark the first step toward realizing versatile all-in-one topological Josephson weak-links using monolayer WTe2 . This article is protected by copyright. All rights reserved.

Searching for Majorana quasiparticles at vortex cores in iron-based superconductors

Abstract The unambiguous detection of the Majorana zero mode (MZM), which is essential for future topological quantum computing, has been a challenge in recent condensed matter experiments. The MZM is expected to emerge at the vortex core of topological superconductors as a zero-energy vortex bound state (ZVBS), amenable to detection using scanning tunneling microscopy/spectroscopy (STM/STS). However, the typical energy resolution of STM/STS has made it challenging to distinguish the MZM from the low-lying trivial vortex bound states. Here, we review the recent high-energy-resolution STM/STS experiments on the vortex cores of Fe(Se,Te), where the MZM is expected to emerge, and the energy of the lowest trivial bound states is reasonably high. Tunneling spectra taken at the vortex cores exhibit a ZVBS well below any possible trivial state, suggesting its MZM origin. However, it should be noted that ZVBS is a necessary but not sufficient condition for the MZM; a qualitative feature unique to the MZM needs to be explored. We discuss the current status and issues in the pursuit of such Majorananess, namely the level sequence of the vortex bound states and the conductance plateau of the ZVBS. We also argue for future experiments to confirm the Majorananess, such as the detection of the doubling of the shot-noise intensity and spin polarization of the MZM.

Experimental quantum simulation of a topologically protected Hadamard gate via braiding Fibonacci anyons

Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e., robust against local disturbances. The topological protection, however, requires complicated lattice models and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC-the Fibonacci anyon system-lacks a physical realization, letalone braiding the non-Abelian anyons. Here, we propose a disk model that can simulate the Fibonacci anyon system and construct the topologically protected logical spaces with the Fibonacci anyons. Via braiding the Fibonacci anyons, we can implement universal quantum gates on the logical space. Our disk model merely requires two physical qubits to realize three Fibonacci anyons at the boundary. By 15 sequential braiding operations, we construct a topologically protected Hadamard gate, which is to date the least-resource requirement for TQC. To showcase, we implement a topological Hadamard gate with two nuclear spin qubits, which reaches fidelity by randomized benchmarking. We further prove by experiment that the logical space and Hadamard gate are topologically protected: local disturbances due to thermal fluctuations result in a global phase only. As a platform-independent proposal, our work is a proof of principle of TQC and paves the way toward fault-tolerant quantum computation.

Trapping Layers Prevent Dopant Segregation and Enable Remote Doping of Templated Self-Assembled InGaAs Nanowires.

Selective area epitaxy is a promising approach to define nanowire networks for topological quantum computing. However, it is challenging to concurrently engineer nanowire morphology, for carrier confinement, and precision doping, to tune carrier density. We report a strategy to promote Si dopant incorporation and suppress dopant diffusion in remote doped InGaAs nanowires templated by GaAs nanomembrane networks. Growth of a dilute AlGaAs layer following doping of the GaAs nanomembrane induces incorporation of Si that otherwise segregates to the growth surface, enabling precise control of the spacing between the Si donors and the undoped InGaAs channel; a simple model captures the influence of Al on the Si incorporation rate. Finite element modeling confirms that a high electron density is produced in the channel.

High temperature spin-triplet topological superconductivity in K2Cr3As3

Spin-triplet superconductors are novel materials capable of harboring Majorana bound states that can be used in topological quantum computing. However, such bulk materials are still rare. Here we review the results that established K2Cr3As3 as a spin-triplet superconductor with transition temperature Tc as high as 6.5 K. We focus on the multiple-phases feature, and its exquisite distance to a ferromagnetic quantum critical point which is likely responsible for the high Tc . We touch on the topological aspects of the superconducting state, and suggest that it is a new route to the technical implementation using a topological spin-triplet superconductor at the highest temperature ever.