The topological phases in one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasi-energy $E=0,π$. These modes are qualitatively different for different boundary conditions used. For two-boundary system with symmetric boundary conditions, the interaction energy between two Majorana bound states can be computed, as in the case of a finite wire. Suggestion of observing these modes are provided.
By using Majorana's stellar representation, we give a clear geometrical interpretation of the topological phases of inversion-symmetric polymerized models by mapping the Bloch states of multi-band systems to Majorana stars on the Bloch sphere. While trajectories of Majorana stars of a filled Bloch band exhibit quite different geometrical structures for topologically trivial and nontrivial phases, we further demonstrate that these structures are uniquely determined by distributions of Majorana stars of two high-symmetrical momentum states, which have different parities for topologically differe…
Majorana zero modes are non-Abelian quasiparticles that emerge on the edges of topological phases of superconductors. Evidence of their presence has been reported in transport measurements on engineered superconducting-based nanostructures. In this manuscript we identify signatures of a topologically protected dynamical manipulation of Majorana zero modes via continuous transport measurement during the manipulation. Specifically, we show that, in a two-terminal geometry, the heat pumped across the terminals at low temperature and voltage bias is characterized by a universal value. We show that…
In this study, we show that quantum walk can describe a Majorana fermion when the coin operator constrained by Lorentz covariance and the initial state satisfies the Majorana condition. The time evolution of a Majorana fermion is demonstrated with the numerical simulations and experimentally runs on a real quantum device provided by IBM Quantum System. To reduce errors due to approximation, we proposed a new efficient way to achieve second order accuracy in the near-term quantum computer without increase the complexity of quantum gate circuitry compared with the first order approximation. We s…
The search for topological superconductors and non-Abelian Majorana modes ranks among the most fascinating topics in condensed matter physics. There now exist several fundamental superconducting phases which host symmetry protected or chiral Majorana modes. The latter, namely the chiral Majorana modes are protected by Chern numbers in even dimensions. Here we propose to observe novel chiral Majorana modes by realizing Fulde-Ferrell-Larkin-Ovchinnikov state, i.e. the pairing density wave (PDW) phase in a Weyl semimetal which breaks time-reversal symmetry. Without symmetry protection, the 3D gap…
We study topological phenomena of quantum walks by implementing a novel protocol that extends the range of accessible properties to the eigenvalues of the walk operator. To this end, we experimentally realise for the first time a split-step quantum walk with decoupling, which allows for investigating the effect of a bulk-boundary while realising only a single bulk configuration. We approximate the symmetry protected edge states with high similarities and read out the phase relative to a reference for all modes. In this way we observe eigenvalues which are distinguished by the presence or absen…
High fidelity quantum information processing requires a combination of fast gates and long-lived quantum memories. In this work, we propose a hybrid architecture, where a parity-protected superconducting qubit is directly coupled to a Majorana qubit, which plays the role of a quantum memory. The superconducting qubit is based upon a $π$-periodic Josephson junction realized with gate-tunable semiconducting wires, where the tunneling of individual Cooper pairs is suppressed. One of the wires additionally contains four Majorana zero modes that define a qubit. We demonstrate that this enables the …
Discrete-time quantum walks allow Floquet topological insulator materials to be explored using controllable systems such as ultracold atoms in optical lattices. By numerical simulations, we study the robustness of topologically protected edge states in the presence of decoherence in one- and two-dimensional discrete-time quantum walks. We also develop a simple analytical model quantifying the robustness of these edge states against either spin or spatial dephasing, predicting an exponential decay of the population of topologically protected edge states. Moreover, we present an experimental pro…
Symmetry-protected topological superconductors (TSCs) can host multiple Majorana zero modes (MZMs) at their edges or vortex cores, while whether the Majorana braiding in such systems is non-Abelian in general remains an open question. Here we uncover in theory the unitary symmetry-protected non-Abelian statisitcs of MZMs and propose the experimental realization. We show that braiding two vortices with each hosting $N$ unitary symmetry-protected MZMs generically reduces to $N$ independent sectors, with each sector braiding two different Majorana modes. This renders the unitary symmetry-protecte…
Majorana zero modes are usually attributed to topological superconductors. We study a class of two-dimensional topologically trivial superconductors without chiral edge modes, which nevertheless host robust Majorana zero modes in topological defects. The construction of this minimal single-band model is facilitated by the Hopf map and the Hopf invariant. This work will stimulate investigations of Majorana zero modes in superconductors in the topologically trivial regime.
We construct a decomposition procedure for converting split-step quantum walks into ordinary quantum walks with alternating coins, and we show that this decomposition enables a feasible linear optical realization of split-step quantum walks by eliminating quantum-control requirements. As salient applications, we show how our scheme will simulate Majorana modes and edge states.
We show how angular momentum conservation can stabilise a symmetry-protected quasi-topological phase of matter supporting Majorana quasi-particles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated b…
Majorana-based quantum computation seeks to encode information non-locally in pairs of Majorana zero modes, thereby isolating qubit states from a local noisy environment. In addition to long coherence times, the attractiveness of Majorana-based quantum computing relies on achieving topologically protected Clifford gates from braiding operations. Recent works have conjectured that mean-field BCS calculations may fail to account for non-universal corrections to the Majorana braiding operations. Such errors would be detrimental to Majorana-based topological quantum computing schemes. In this work…
Quantum information protected by the topology of the storage medium is expected to exhibit long coherence times. Another feature are topologically protected gates generated through braiding of Majorana bound states. However, braiding requires structures with branched topological segments which have inherent difficulties in the semiconductor-superconductor heterostructures now believed to host Majorana bound states. In this paper, we construct quantum bits taking advantage of the topological protection and non-local properties of Majorana bound states in a network of parallel wires, but without…
Floquet topological matter has emerged as one exciting platform to explore rich physics and game-changing applications of topological phases. As one remarkable and recently discovered feature of Floquet symmetry protected topological (SPT) phases, in principle a simple periodically driven system can host an arbitrary number of topological protected zero edge modes and pi edge modes, with Majorana zero modes and Majorana pi modes as examples protected by the particle-hole symmetry. This work advocates a new route to holonomic quantum computation by exploiting the co-existence of many Floquet SP…
We propose a method of encoding a topologically-protected qubit using Majorana fermions in a trapped-ion chain. This qubit is protected against major sources of decoherence, while local operations and measurements can be realized. Furthermore, we show that an efficient quantum interface and memory for arbitrary multiqubit photonic states can be built, encoding them into a set of entangled Majorana-fermion qubits inside cavities.
We present designs for scalable quantum computers composed of qubits encoded in aggregates of four or more Majorana zero modes, realized at the ends of topological superconducting wire segments that are assembled into superconducting islands with significant charging energy. Quantum information can be manipulated according to a measurement-only protocol, which is facilitated by tunable couplings between Majorana zero modes and nearby semiconductor quantum dots. Our proposed architecture designs have the following principal virtues: (1) the magnetic field can be aligned in the direction of all …
We calculate the topological boundary modes found for quantum wire for a quantum ring. For the symmetric ring we find analytical solutions for two quasi-particles identifiable as the Majorana states and for asymmetric ring, we have also find approximate solution. By turning on a small field at time zero, we find a field induced oscillation of the Majorana states that was initially localized. The range of validity of our solution, the relation between the period of oscillation and the coin parameter and field strength, and the encoding of a message of the on/off of the field using the Majorana …
Discrete time crystals have attracted considerable theoretical and experimental studies but their potential applications have remained unexplored. A particular type of discrete time crystals, termed "Majorana time crystals", is found to emerge in a periodically driven superconducting wire accommodating two different species of topological edge modes. It is further shown that different Majorana edge modes separated in the time lattice can be braided, giving rise to an unforeseen scenario for topologically protected gate operations. The proposed braiding scheme can also generate a magic state th…
Because Majorana zero modes store quantum information non-locally, they are protected from noise, and have been proposed as a building block for a quantum computer. We show how to use the same protection from noise to implement universal fermionic quantum computation. Our architecture requires only two Majoranas to encode a fermionic quantum degree of freedom, compared to alternative implementations which require a minimum of four Majoranas for a spin quantum degree of freedom. The fermionic degrees of freedom support both unitary coupled cluster variational quantum eigensolver and quantum pha…
Localized Majorana fermions emerge in many topologically ordered systems and exhibit exchange statistics of Ising anyons. This enables noise-resistant implementation of a limited set of operations by braiding and fusing Majorana fermions. Unfortunately, these operations are incapable of implementing universal quantum computation. We show that, regardless of these limitations, Majorana fermions could be used to demonstrate non-locality (correlations incompatible with a local hidden variable theory) in experiments using only topologically protected operations. We also demonstrate that our propos…
Majorana modes, typically arising at the edges of one-dimensional topological superconductors, are considered to be a promising candidate for encoding nonlocal qubits in fault-tolerant quantum computation. Here we propose to exploit the two-dimensional geometry of Majorana corner modes in second-order topological superconductors to realize measurement-only quantum computation. In particular, a periodically modulated topological superconducting system may host eight Majorana corner modes, through which two logical qubits as well as one ancilla qubit can be constructed and topologically protecte…
This paper uncovers and exploits a link between a central object in harmonic analysis, the so-called Schur functions, and the very hot topic of symmetry protected topological phases of quantum matter. This connection is found in the setting of quantum walks, i.e. quantum analogs of classical random walks. We prove that topological indices classifying symmetry protected topological phases of quantum walks are encoded by matrix Schur functions built out of the walk. This main result of the paper reduces the calculation of these topological indices to a linear algebra problem: calculating symmetr…
Linear-optical systems can implement photonic quantum walks that simulate systems with nontrivial topological properties. Here, such photonic walks are used to jointly entangle polarization and winding number. This joint entanglement allows information processing tasks to be performed with interactive access to a wide variety of topological features. Topological considerations are used to suppress errors, with polarization allowing easy measurement and manipulation of qubits. We provide three examples of this approach: production of two-photon systems with entangled winding number (including t…
In this paper we study the symmetry protected Majorana edge states for the Z2 topological order of the Wen-plaquette model and the toric-code model and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at k=0 and k=pi. For the edge states of the toric-code model without translational symmetry, the edge modes become gapped.
A universal quantum computer requires a full set of basic quantum gates. With Majorana bound states one can form all necessary quantum gates in a topologically protected way, bar one. In this manuscript we present a protocol that achieves the missing, so called, $π/8$ 'magic' phase gate. The protocol is based on the manipulation of geometric phases in a universal manner, and does not require fine tuning for distinct physical realizations. The protocol converges exponentially with the number of steps in the geometric path. Furthermore, the magic gate protocol relies on the most basic hardware p…
Among the major approaches that are being pursued for realizing quantum bits, the Majorana-based platform has been the most recent to be launched. It attempts to realize qubits which store quantum information in a topologically-protected manner. The quantum information is protected by its nonlocal storage in localized and well-separated Majorana zero modes, and manipulated by exploiting their nonabelian quantum exchange properties. Realizing these topological qubits is experimentally challenging, requiring superconductivity, helical electrons (created by spin-orbit coupling) and breaking of ti…
Recently, effects of nonlinearity on topologically nontrivial systems have attracted attention and the stability of topologically protected edge states has been studied for a quantum walk with nonlinear effects, which is akin to time-periodically driven systems (Floquet systems). In the previous work, it has been found that the edge states can be stable attractors or unstable repellers depending on their intrinsic topological property, while the stability is not affected by the strength of nonlinearity. In the present work, we find additional bifurcations at which edge states change from stabl…
Topological excitations, such as Majorana zero modes, are a promising route for encoding quantum information. Topologically protected gates of Majorana qubits, based on their braiding, will require some form of network. Here, we propose to build such a network by entangling Majorana matter with light in a microwave cavity QED setup. Our scheme exploits a light-induced interaction which is universal to all the Majorana nanoscale circuit platforms. This effect stems from a parametric drive of the light-matter coupling in a one-dimensional chain of physical Majorana modes. Our setup enables all t…
The evolution of quantum walk on a finite wire under a small increment of vector potential $α$ can exhibit intrinsic quantum oscillation of the two topologically protected bound states corresponding to the Majorana modes. By tuning an external electric field corresponding to the addition of an $α$ impulse at the end of each intrinsic period, the intrinsic oscillation is enveloped by a beat modulation with a longer period. This beat oscillation is useful in the preparation of decoherence-free qubit in trapped ion chain and may be observed in several experiments.
Qubits constructed from uncoupled Majorana fermions are protected from decoherence, but to perform a quantum computation this topological protection needs to be broken. Parity-protected quantum computation breaks the protection in a minimally invasive way, by coupling directly to the fermion parity of the system --- irrespective of any quasiparticle excitations. Here we propose to use a superconducting charge qubit in a transmission line resonator (a socalled transmon) to perform parity-protected rotations and read-out of a topological (top) qubit. The advantage over an earlier proposal using …
The CNOT gate is a two-qubit gate which is essential for universal quantum computation. A well-established approach to implement it within Majorana-based qubits relies on subsequent measurement of (joint) Majorana parities. We propose an alternative scheme which operates a protected CNOT gate via the holonomic control of a handful of system parameters, without requiring any measurement. We show how the adiabatic tuning of pair-wise couplings between Majoranas can robustly lead to the full entanglement of two qubits, insensitive with respect to small variations in the control of the parameters.
Majorana bosons, that is, tight bosonic analogs of the Majorana fermionic quasiparticles of condensed matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and nontrivial bulk topology makes their emergence possible in noninteracting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability, whereby a conserved Majorana boson localized on one edge is paired, in general, with a symmetry generator localized on the opposite edge. We argue th…
Braiding of Majorana zero modes provides a promising platform for quantum information processing, which is topologically protected against errors. Strictly speaking, however, the scheme relies on infinite braiding times as it utilizes the adiabatic limit. Here we show how to minimize nonadiabatic errors for finite braiding times by finding an optimal protocol for the Majorana movement. Interestingly, these protocols are characterized by sharp transitions between Majorana motion at maximal and minimal velocities. We find that these so-called bang-bang protocols can minimize the nonadiabatic tra…
We have studied the low-lying excitations of a chain of coupled circuit-QED systems, and report several intriguing properties of its two nearly degenerate ground states. The ground states are Schrödinger cat states at a truly large scale, involving maximal entanglement between the resonator and the qubit, and are mathematically equivalent to Majorana bound states. With a suitable design of physical qubits, they are protected against local fluctuations and constitute a non-local qubit. Further, they can be probed and manipulated coherently by attaching an empty resonator to one end of the circu…
Coupling the surface state of a topological insulator (TI) to an s-wave superconductor is predicted to produce the long-sought Majorana quasiparticle excitations. However, superconductivity has not been measured in surface states when the bulk charge carriers are fully depleted, i.e., in the true topological regime that is relevant for investigating Majorana modes. Here, we report measurements of DC Josephson effects in TI-superconductor junctions as the chemical potential is moved from the bulk bands into the band gap, or through the true topological regime characterized by the presence of on…
Majorana fermions hold promise for quantum computation, because their non-Abelian braiding statistics allows for topologically protected operations on quantum information. Topological qubits can be constructed from pairs of well-separated Majoranas in networks of nanowires. The coupling to a superconducting charge qubit in a transmission line resonator (transmon) permits braiding of Majoranas by external variation of magnetic fluxes. We show that readout operations can also be fully flux-controlled, without requiring microscopic control over tunnel couplings. We identify the minimal circuit th…
Surface codes offer a very promising avenue towards fault-tolerant quantum computation. We argue that two-dimensional interacting networks of Majorana bound states in topological superconductor/semiconductor heterostructures hold several distinct advantages in that direction, both concerning the hardware realization and the actual operation of the code. We here discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. All physical ingredients needed to implement these operations are routinely used in to…
Universal quantum computation (UQC) using Majorana fermions on a 2D topological superconducting (TS) medium remains an outstanding open problem. This is because the quantum gate set that can be generated by braiding of the Majorana fermions does not include \emph{any} two-qubit gate and also the single-qubit $π/8$ phase gate. In principle, it is possible to create these crucial extra gates using quantum interference of Majorana fermion currents. However, it is not clear if the motion of the various order parameter defects (vortices, domain walls, \emph{etc.}), to which the Majorana fermions ar…
We demonstrate how the non-trivial interplay between spin-orbit coupling and nodeless $s$-wave superconductivity can drive a fully gapped two-band topological insulator into a time-reversal invariant gapless topological superconductor supporting symmetry-protected Majorana flat bands. We characterize topological phase diagrams by a ${\mathbb Z}_2 \times{\mathbb Z}_2$ partial Berry-phase invariant, and show that, despite the trivial crystal geometry, no unique bulk-boundary correspondence exists. We trace this behavior to the anisotropic quasiparticle bulk gap closing, linear vs. quadratic, and…
The oscillation of Majorana modes with near zero energy plays a very important role for ascertaining Majorana fermions. The edge states, which also have almost-zero-energy in one-dimensional Su-Schrieffer-Heeger chain (SSHc), have been extensively studied for their topologically protected properties when the on-sites have dissipations induced by independent environments. We here show that common environments shared by each pair of the nearest neighbour sites in the SSHc can result in dissipative couplings between sites, and thus change topologically trivial phase to nontrivial one. The Majoran…
Motivated by recent experimental progress towards the measurement and manipulation of Majorana fermions with superconducting circuits, we propose a device interfacing Majorana fermions with circuit quantum electrodynamics. The proposed circuit acts as a charge parity detector changing the resonance frequency of a superconducting $λ/4$ - resonator conditioned on the parity of charges on nearby gates. Operating at both charge and flux sweet spots, this device is highly insensitive to environmental noise. It enables high-fidelity single-shot quantum non demolition readout of the state of a pair o…
Majorana zero modes (MZMs) promise a platform for topologically protected fermionic quantum computation. However, creating multiple MZMs and generating (directly or via measurements) the requisite transformations (e.g., braids) pose significant challenges. We introduce fermion-parity-based computation (FPBC): a measurement-based scheme, modeled on Pauli-based computation, that uses efficient classical processing to virtually increase the number of available MZMs and which, given magic state inputs, operates without transformations. FPBC requires all MZM parities to be measurable, but this conf…
We introduce two-dimensional topological insulators in proximity to high-temperature cuprate or iron-based superconductors as high-temperature platforms of Majorana Kramers pairs of zero modes. The proximity-induced pairing at the helical edge state of the topological insulator serves as a Dirac mass, whose sign changes at the sample corner because of the pairing symmetry of high-$T_c$ superconductors. This sign changing naturally creates at each corner a pair of Majorana zero modes protected by time-reversal symmetry. Conceptually, this is a topologically trivial superconductor-based approach…
We propose a three-terminal structure to probe robust signatures of Majorana zero modes. This structure consists of a quantum dot coupled to the normal metal, s-wave superconducting and Majorana Y-junction leads. The zero-bias differential conductance at zero temperature of the normal-metal lead peaks at $2e^{2}/h$, which will be deflected after Majorana braiding. This quantized conductance can entirely arise from the Majorana-induced crossed Andreev reflection, protected by the energy gap of the superconducting lead. We find that the effect of thermal broadening is significantly suppressed wh…
Electrons confined in Si quantum dots possess orbital, spin, and valley degrees of freedom (d.o.f.). We perform Landau-Zener-Stuckelberg-Majorana (LZSM) interferometry on a Si double quantum dot that is strongly coupled to a microwave cavity to probe the valley d.o.f. The resulting LZSM interference pattern is asymmetric as a function of level detuning and persists for drive periods that are much longer than typical charge decoherence times. By correlating the LZSM interference pattern with charge noise measurements, we show that valley-orbit hybridization provides some protection from the del…
We present quantum control techniques to engineer flat bands of symmetry-protected Majorana edge modes in s-wave superconductors. Specifically, we show how periodic control may be employed for designing time-independent effective Hamiltonians, which support $Floquet$ $Majorana$ $flat$ $bands$, starting from equilibrium conditions that are either topologically trivial or only support individual Majorana pairs. In the first approach, a suitable modulation of the chemical potential simultaneously induces Majorana flat bands and dynamically $activates$ a pre-existing chiral symmetry which is respo…
Schemes for topological quantum computation with Majorana bound states rely heavily on the ability to measure products of Majorana operators projectively. Here, we employ Markovian quantum measurement theory, including the readout device, to analyze such measurements. Specifically, we focus on the readout of Majorana qubits via continuous charge sensing of a tunnel-coupled quantum dot by a quantum point contact. We show that projective measurements of Majorana products $\prod_i\hatγ_i$ can be implemented by continuous charge sensing under quite general circumstances. Essential requirements are…
Exotic topologically protected zero modes with parafermionic statistics (also called fractionalized Majorana modes) have been proposed to emerge in devices fabricated from a fractional quantum Hall system and a superconductor. The fractionalized statistics of these modes takes them an important step beyond the simplest non-Abelian anyons, Majorana fermions. Building on recent advances towards the realization of fractional quantum Hall states of bosonic ultracold atoms, we propose a realization of parafermions in a system consisting of Bose-Einstein-condensate trenches within a bosonic fraction…
We aim to study a one-dimensional $p$-wave superconductor with quasiperiodic on-site potentials. A modified real-space-Pfaffian method is applied to calculate the topological invariants. We confirm that the Majorana zero mode is protected by the nontrivial topology the topological phase transition is accompanied by the energy gap closing and reopening. In addition, we numerically find that there are mobility edges which originate from the competition between the extended $p$-wave pairing and the localized quasi-disorder. We qualitatively analyze the influence of superconducting pairing paramet…
Majorana zero modes (MZM-s) occurring at the edges of a 1D, p-wave, spinless superconductor, in absence of fluctuations of the phase of the superconducting order parameter, are quintessential examples of topologically-protected zero-energy modes occurring at the edges of 1D symmetry-protected topological phases. In this work, we numerically investigate the fate of the topological phase in the presence of phase-fluctuations using the density matrix renormalization group (DMRG) technique. To that end, we consider a one-dimensional array of MZM-s on mesoscopic superconducting islands at zero temp…
We investigate the resilience of symmetry-protected topological edge states at the boundaries of Kitaev chains in the presence of a bath which explicitly introduces symmetry-breaking terms. Specifically, we focus on single-particle losses and gains, violating the protecting parity symmetry, which could generically occur in realistic scenarios. For homogeneous systems, we show that the Majorana mode decays exponentially fast. By the inclusion of strong disorder, where the closed system enters a many-body localized phase, we find that the Majorana mode can be stabilized substantially. The decay …
