Topological Quantum Computation Research Articles

Topological surface superconductivity in FeSe0.45Te0.55

The engineering of Majorana zero modes in topological superconductors, a paradigm for the realization of topological quantum computing and topology-based devices, has been hampered by the absence of materials with sufficiently large superconducting gaps. Recent experiments, however, have provided enthralling evidence for the existence of topological surface superconductivity in the iron-based superconductor FeSe0.45Te0.55 possessing a full s±-wave gap of a few meV. Here, we propose a mechanism for the emergence of topological superconductivity on the surface of FeSe0.45Te0.55 by demonstrating that the interplay between the s±-wave symmetry of the superconducting gap, surface magnetism, and a Rashba spin–orbit interaction gives rise to robust topological superconducting phases. Moreover, the proposed mechanism explains a series of experimentally observed hallmarks of topological superconductivity, such as the emergence of Majorana zero modes in the center of vortex cores and at the end of line defects, as well as of chiral Majorana edge modes along domain walls. We also propose that the spatial distribution of supercurrents near a domain wall is a characteristic signature measurable via a scanning superconducting quantum interference device that can distinguish between chiral Majorana edge modes and trivial in-gap states.

Quantum teleportation using Ising anyons

Anyons have been extensively investigated as information carriers in topological quantum computation. However, how to characterize the information flow in quantum networks composed of anyons is less understood, which motivates us to study quantum communication protocols in anyonic systems. Here we propose a general topologically protected protocol for quantum teleportation based on the Ising anyon model and prove that with our protocol an unknown anyonic state of any number of Ising anyons can be teleported from Alice to Bob. Our protocol naturally generalizes quantum state teleportation from systems of locally distinguishable particles to systems of Ising anyons, which may promote our understandings of anyonic quantum entanglement as a quantum resource. In addition, our protocol is expected to be realized with the Majorana zero modes, one of the possible physical realizations for the Ising anyon in experiments.

High-Throughput Investigations of Topological and Nodal Superconductors.

The theory of symmetry indicators has enabled database searches for topological materials in normal conducting phases, which has led to several encyclopedic topological material databases. To date, such a database for topological superconductors is yet to be achieved because of the lack of information about pairing symmetries of realistic materials. In this Letter, sidestepping this issue, we tackle an alternative problem: the predictions of topological and nodal superconductivity in materials for each single-valued representation of point groups. Based on recently developed symmetry indicators for superconductors, we provide comprehensive mappings from pairing symmetries to the topological or nodal superconducting nature for nonmagnetic materials listed in the Inorganic Crystal Structure Database. We quantitatively show that around 90% of computed materials are topological or nodal superconductors when a pairing that belongs to a one-dimensional nontrivial representation of point groups is assumed. When materials are representation-enforced nodal superconductors, positions and shapes of the nodes are also identified. When combined with experiments, our results will help us understand the pairing mechanism and facilitate realizations of the long-sought Majorana fermions promising for topological quantum computations.

Extend the Levin-Wen model to two-dimensional topological orders with gapped boundary junctions

A realistic material may possess defects, which often bring the material new properties that have practical applications. The boundary defects of a two-dimensional topologically ordered system are thought of as an alternative way of realizing topological quantum computation. To facilitate the study of such boundary defects, in this paper, we construct an exactly solvable Hamiltonian model of topological orders with gapped boundary junctions, where the boundary defects reside, by placing the Levin-Wen model on a disk, whose gapped boundary is separated into multiple segments by junctions. We derive a formula of the ground state degeneracy and an explicit ground-state basis of our model. We propose the notion of mobile and immobile charges on the boundary and find that they are quantum observables and label the ground-state basis. Our model is computation friendly.

A new type of cyclotron resonance from charge-impurity scattering in the bulk-insulating Bi2Se3 thin films

This work focuses on the low frequency Drude response of bulk-insulating topological insulator (TI) Bi2Se3 films. The frequency and field dependence of the mobility and carrier density are measured simultaneously via time-domain terahertz spectroscopy. These films are grown on buffer layers, capped by Se, and have been exposed in air for months. Under a magnetic field up to 7 Tesla, we observe prominent cyclotron resonances (CRs). We attribute the sharp CR to two different topological surface states from both surfaces of the films. The CR sharpens at high fields due to an electron-impurity scattering. By using magneto-terahertz spectroscopy, we confirm that these films are bulk-insulating, which paves the way to use intrinsic TIs without bulk carriers for applications including topological spintronics and quantum computing.

Dual topological states in the layered titanium-based oxypnictide superconductor BaTi2Sb2O

Topological superconductors have long been predicted to host Majorana zero modes which obey non-Abelian statistics and have potential for realizing non-decoherence topological quantum computation. However, material realization of topological superconductors is still a challenge in condensed matter physics. Utilizing high-resolution angle-resolved photoemission spectroscopy and first-principles calculations, we predict and then unveil the coexistence of topological Dirac semimetal and topological insulator states in the vicinity of Fermi energy (EF) in the titanium-based oxypnictide superconductor BaTi2Sb2O. Further spin-resolved measurements confirm its spin-helical surface states around EF, which are topologically protected and give an opportunity for realization of Majorana zero modes and Majorana flat bands in one material. Hosting dual topological states, the intrinsic superconductor BaTi2Sb2O is expected to be a promising platform for further investigation of topological superconductivity.

Steering Majorana braiding via skyrmion-vortex pairs: A scalable platform

Majorana zero modes are quasiparticles that hold promise as building blocks for topological quantum computing. However, the litmus test for their detection, the observation of exotic non-abelian statistics revealed by braiding, has so far eluded experimental efforts. Here we take advantage of the fact that skyrmion-vortex pairs in superconductor-ferromagnet heterostructures harboring Majorana zero modes can be easily manipulated in two spatial dimensions. We adiabatically braid the hybrid topological structures and explicitly confirm the non-abelian statistics of the Majorana zero modes numerically using a self-consistent calculation of the superconducting order parameter. Our proposal of controlling skyrmion-vortex pairs provides the necessary leeway toward a scalable topological quantum computing platform.

Ordered and tunable Majorana-zero-mode lattice in naturally strained LiFeAs.

Majorana zero modes (MZMs) obey non-Abelian statistics and are considered building blocks for constructing topological qubits1,2. Iron-based superconductors with topological bandstructures have emerged as promising hosting materials, because isolated candidate MZMs in the quantum limit have been observed inside the topological vortex cores3-9. However, these materials suffer from issues related to alloying induced disorder, uncontrolled vortex lattices10-13 and a low yield of topological vortices5-8. Here we report the formation of an ordered and tunable MZM lattice in naturally strained stoichiometric LiFeAs by scanning tunnelling microscopy/spectroscopy. We observe biaxial charge density wave (CDW) stripes along the Fe-Fe and As-As directions in the strained regions. The vortices are pinned on the CDW stripes in the As-As direction and form an ordered lattice. We detect that more than 90 per cent of the vortices are topological and possess the characteristics of isolated MZMs at the vortex centre, forming an ordered MZM lattice with the density and the geometry tunable by an external magnetic field. Notably, with decreasing the spacing of neighbouring vortices, the MZMs start to couple with each other. Our findings provide a pathway towards tunable and ordered MZM lattices as a platform for future topological quantum computation.

Prediction of new topological superconductor candidate 1T−Ti(Se1−yTey)2

Topological superconductivity has attracted intensive interest for its ability of hosting Majorana zero mode and implementing in topological quantum computations. Based on the first-principles calculations and the analysis of the effective BdG Hamiltonian, we demonstrate that 1$T$-TiTe$_2$ is a topological metal hosting Dirac cone type of surface states near the Fermi level, and it exhibits a normal-topological-normal superconductivity phase transition as a function of the chemical potential. These results point out a new promising topological superconductor without random dopant, in which the influence of the impurity may be greatly reduced. Furthermore, our calculations also suggest that the transition metal intercalated Ti(Se$_{1-y}$Te$_y$)$_2$ is also a highly possible route to realize TSC and MZMs.

Edge Z3 parafermions in fermionic lattices

Parafermions modes are non-Abelian anyons which were introduced as $\mathbb{Z}_N$ generalizations of $\mathbb{Z}_2$ Majorana states. In particular, $\mathbb{Z}_3$ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of theoretical work on $\mathbb{Z}_3$ parafermions has relied on bosonization methods or parafermionic quasi-particles. In this work, we introduce a representation of $\mathbb{Z}_3$ parafermions in terms of purely fermionic models operators in the t-J regime. We establish the equivalency of a family of lattice fermionic models written in the $t-J$ model basis with a Kitaev-like chain supporting free $\mathbb{Z}_3$ parafermonic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing $\mathbb{Z}_3$ parafermions in strongly interacting electronic systems.

Coexistence of pressure-induced superconductivity and topological surface states in elementary substance Sb

Topological superconductors are expected to host non-Abelian Majorana zero mode which is of fundamental importance for topological quantum computing. However, the discovered topological superconductor candidates are still rare. Here, we systematically studied the topological semimetal antimony Sb through first-principles calculations, and found that a moderate pressure could induce the superconductivity coexisting with topological surface states. The superconducting critical transition temperature (${T}_{c}$) reaches $\ensuremath{\sim}0.3\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ under $\ensuremath{\sim}8\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ and the topological surface states can be well preserved. The ${T}_{c}$ can be much enhanced to $\ensuremath{\sim}2\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ through further electron doping. Elementary substance antimony may provide a simple material platform to detect the Majorana boundary states and develop the topological quantum computation.

Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

It is shown that the representation theory of some finitely presented groups thanks to their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface fH (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers E˜6 and D˜4 contain fH. A surface birationally equivalent to a K3 surface is another compound of their character varieties.

Majorana-like end states in one-dimensional dimerized Kitaev topolectrical circuit

Majorana zero modes (MZMs) have attracted tremendous attention in condensed matter and materials physics communities due to the implications in topological quantum computation. One-dimensional (1D) dimerized Kitaev chain is a prototype model for MZMs, but its realization remains a challenge in material systems. Here, we develop a distinctive approach to achieve Majorana-like end states (MESs) by implementing practical dimerized Kitaev topolectrical circuits. Specifically, two arrays of inductors are arranged to simulate particles and antiparticles, while intra- and inter-array capacitive connections are used to model hopping and superconducting pairing. Three topological phases can be achieved by tuning the capacitance, i.e. the trivial phase, Su–Schrieffer–Heeger topological phase and Kitaev phase, with distinct field strength distributions in real space. Majorana splitting is observed around a domain wall in the circuit, and we propose an efficient experimental observable-edge distance-to characterize the process as premonition of topological phase transition. Remarkably, dynamics of the Gaussian wave packet in time domain provide an excellent signal to detect MESs in experiments, as only MESs allow nonlocal propagation in circuit network. Our results not only manifest the superiorities of topolectrical circuits for exotic topological states, but also pave the way for possible applications in electrical engineering and signal processing.

Universal topological quantum computation with strongly correlated Majorana edge modes

Majorana-based quantum gates are not complete for performing universal topological quantum computation while Fibonacci-based gates are difficult to be realized electronically and hardly coincide with the conventional quantum circuit models. In reference Hu and Kane (2018 Phys. Rev. Lett. 120 066801), it has been shown that a strongly correlated Majorana edge mode in a chiral topological superconductor can be decomposed into a Fibonacci anyon τ and a thermal operator anyon ɛ in the tricritical Ising model. The deconfinement of τ and ɛ via the interaction between the fermion modes yields the anyon collisions and gives the braiding of either τ or ɛ. With these braidings, the complete members of a set of universal gates, the Pauli gates, the Hadamard gate and extra phase gates for one-qubit as well as controlled-NOT (CNOT) gate for two-qubits, are topologically assembled. Encoding quantum information and reading out the computation results can be carried out through electric signals. With the sparse-dense mixed encodings, we set up the quantum circuit where the CNOT gate turns out to be a probabilistic gate and design the corresponding devices with thin films of the chiral topological superconductor. As an example of the universal topological quantum computing, we show the application to Shor’s integer factorization algorithm.

Fusion of Majorana bound states with mini-gate control in two-dimensional systems

A hallmark of topological superconductivity is the non-Abelian statistics of Majorana bound states (MBS), its chargeless zero-energy emergent quasiparticles. The resulting fractionalization of a single electron, stored nonlocally as a two spatially-separated MBS, provides a powerful platform for implementing fault-tolerant topological quantum computing. However, despite intensive efforts, experimental support for MBS remains indirect and does not probe their non-Abelian statistics. Here we propose how to overcome this obstacle in mini-gate controlled planar Josephson junctions (JJs) and demonstrate non-Abelian statistics through MBS fusion, detected by charge sensing using a quantum point contact, based on dynamical simulations. The feasibility of preparing, manipulating, and fusing MBS in two-dimensional (2D) systems is supported in our experiments which demonstrate the gate control of topological transition and superconducting properties with five mini gates in InAs/Al-based JJs. While we focus on this well-established platform, where the topological superconductivity was already experimentally detected, our proposal to identify elusive non-Abelian statistics motivates also further MBS studies in other gate-controlled 2D systems.

Density functional Bogoliubov-de Gennes analysis of superconducting Nb and Nb(110) surfaces

We report on the implementation of the Bogoliubov-de Gennes method into the JuKKR code, an implementation of the relativistic all-electron, full-potential Korringa-Kohn-Rostoker Green function method, which allows a material-specific description of inhomogeneous superconductors and heterostructures on the basis of density functional theory. We describe the formalism and report on calculations for the $s$-wave superconductor Nb, a potential component of the materials platform enabling the realization of the Majorana zero modes in the field of topological quantum computing. We compare the properties of the superconducting state both in the bulk and for (110) surfaces. We compare slab calculations for different thicknesses, the effect of surface relaxations, and the influence of a softening of phonon modes on the surface for the resulting superconducting gap.

Robustness of majorana zero-energy state

Based on the principle of linearized stability proposed by Lyapunov, we investigate the robustness of Majorana zero-energy state (MZES), which plays an important role in topological quantum computation. We show that the MZES is not robust enough against the external perturbations, because mathematically it is critically stable instead of being asymptotically stable, and only the states with asymptotic stability can be regarded as robust, therefore, the MZES cannot be used to carry quantum information in topological quantum computation. Our study is different from previous works that usually make the numerical tests by some special perturbations, whereas our analytical derivation is suitable for arbitrary perturbations. As an example, we demonstrate this by the stability analysis of MZES in the spin-orbit-coupled semiconductor/superconductor junction.

Identifying Abelian and non-Abelian topological orders in the string-net model using a quantum scattering circuit

Realizing universal topological quantum computers requires the manipulation of non-Abelian topological orders in a physical system, which presents great challenges. Conversely, the rapid development in circuit-based quantum computing offers a reliable quantum simulation approach to study these topological orders. The preliminary problem is how to identify distinct topological orders. Here, we develop a framework based on the quantum scattering circuit to directly and efficiently measure the modular transformation matrix, which is widely deemed as the fingerprint of a given topological order. The information of the modular transformation matrix is encoded in the probe qubit, and the readout merely requires single-qubit Pauli measurements. We further implement the scheme in a nuclear magnetic resonance quantum simulator to emulate the string-net model, where an Abelian ${\mathbb{Z}}_{2}$ toric code and a non-Abelian Fibonacci order emerge. In particular, the latter is predicted to be the simplest candidate for universal topological quantum computers. The two topological orders are unambiguously distinguished by the experimentally measured modular transformation matrices. As an experimental demonstration of a non-Abelian topological order with efficient readout, our work may open avenues toward investigating topological orders in circuit-based quantum simulators.

Quantum walks of two correlated photons in a 2D synthetic lattice

Quantum walks represent paradigmatic quantum evolutions, enabling powerful applications in the context of topological physics and quantum computation. They have been implemented in diverse photonic architectures, but the realization of two-particle dynamics on a multidimensional lattice has hitherto been limited to continuous-time evolutions. To fully exploit the computational capabilities of quantum interference it is crucial to develop platforms handling multiple photons that propagate across multidimensional lattices. Here, we report a discrete-time quantum walk of two correlated photons in a two-dimensional lattice, synthetically engineered by manipulating a set of optical modes carrying quantized amounts of transverse momentum. Mode-couplings are introduced via the polarization-controlled diffractive action of thin geometric-phase optical elements. The entire platform is compact, efficient, scalable, and represents a versatile tool to simulate quantum evolutions on complex lattices. We expect that it will have a strong impact on diverse fields such as quantum state engineering, topological quantum photonics, and Boson Sampling.

Selective area epitaxy of PbTe-Pb hybrid nanowires on a lattice-matched substrate

Topological quantum computing is based on braiding of Majorana zero modes encoding topological qubits. A promising candidate platform for Majorana zero modes is semiconductor-superconductor hybrid nanowires. The realization of topological qubits and braiding operations requires scalable and disorder-free nanowire networks. Selective area growth of in-plane InAs and InSb nanowires, together with shadow-wall growth of superconductor structures, have demonstrated this scalability by achieving various network structures. However, the noticeable lattice mismatch at the nanowire-substrate interface, acting as a disorder source, imposes a serious obstacle along with this roadmap. Here, combining selective area and shadow-wall growth, we demonstrate the fabrication of PbTe-Pb hybrid nanowires - another potentially promising Majorana system - on a nearly perfectly lattice-matched substrate CdTe, all done in one molecular beam epitaxy chamber. Transmission electron microscopy shows the single-crystal nature of the PbTe nanowire and its atomically sharp and clean interfaces to the CdTe substrate and the Pb overlayer, without noticeable inter-diffusion or strain. The nearly ideal interface condition, together with the strong screening of charge impurities due to the large dielectric constant of PbTe, hold promise towards a clean nanowire system to study Majorana zero modes and topological quantum computing.