Topological quantum computation away from the ground state using Majorana fermions

Topological quantum computation provides an elegant way around decoherence, as one encodes quantum information in a non-local fashion that the environment finds difficult to corrupt. Here we establish that one of the key operations---braiding of non-Abelian anyons---can be implemented in one-dimensional semiconductor wire networks. Previous work [Lutchyn et al., arXiv:1002.4033 and Oreg et al., arXiv:1003.1145] provided a recipe for driving semiconducting wires into a topological phase supporting long-sought particles known as Majorana fermions that can store topologically protected quantum information. Majorana fermions in this setting can be transported, created, and fused by applying locally tunable gates to the wire. More importantly, we show that networks of such wires allow braiding of Majorana fermions and that they exhibit non-Abelian statistics like vortices in a p+ip superconductor. We propose experimental setups that enable the Majorana fusion rules to be probed, along with networks that allow for efficient exchange of arbitrary numbers of Majorana fermions. This work paves a new path forward in topological quantum computation that benefits from physical transparency and experimental realism.

Verifying non-Abelian statistics by numerical braiding Majorana fermions

Majorana fermions in condensed matter: An outlook

: Spin has long been known to have the potential to perform universal quantum computation. To realize quantum computation with spins one needs an extraordinary amount of control over the spins so we can set, manipulate, and read out the various processes required in information processing. This is a daunting task as these processes are not immune from decoherence. While one can manipulate single spins with high-fidelity and long coherence times, there is no clear path to move from single spin manipulation to many spins. Recent advances have revealed a new type of information processing, topological quantum computation, which is immune from environment related decoherence. Topological information processing relies on the manipulation of anyons, particles which obey non-Abelian statistics. The simplest of these particles, Majorana fermions, are believed to exist as excitations in exotic materials under extreme conditions. Additionally, Majorana fermions have been proposed to exist in a new class of materials commonly referred to as topological insulators coupled with superconducting contacts under much less extreme conditions. In this work, we propose to theoretically investigate the formation, manipulation, entanglement and detection of Majorana fermions in diamond-topological insulator-superconductor heterojunctions. Furthermore, we propose to further investigate the relationship between superconducting elements and 2D and 3D topological insulators. The tasks we outline in this proposal will be undertaken using a variety of analytical tools to provide insight into the system behavior at the theoretical limits and numerical techniques to yield results in experimentally realistic conditions. The goal is to make significant and measurable advances towards transformative information processing technologies that have the potential to vastly increase operational capabilities of multiple Air Force applications.

Majorana bound states are interesting candidates for applications in topological quantum computation. Low energy models allowing to grasp their properties are hence conceptually important. The usual scenario in these models is that two relevant gapped phases, separated by a gapless point, exist. In one of the phases, topological boundary states are absent, while the other one supports Majorana bound states. We show that a customary model violates this paradigm. The phase that should not host Majorana fermions supports a fractional soliton exponentially localized at only one end. By varying the parameters of the model, we describe analytically the transition between the fractional soliton and two Majorana fermions. Moreover, we provide a possible physical implementation of the model. We further characterize the symmetry of the superconducting pairing, showing that the odd-frequency component is intimately related to the spatial profile of the Majorana wavefunctions.

One-dimensional time-reversal invariant topological superconducting wires of the symmetry class DIII exhibit exotic physics which can be exploited to realize the set of universal operations in topological quantum computing. However, the verification of DIII-class physics in conventional condensed matter materials is highly nontrivial due to realistic constraints. Here we propose a symmetry-protected hard-core boson simulator of the one-dimensional DIII topological superconductor. By using the developed dispersive dynamic modulation approach, not only the faithful simulation of this new type of spinful superconducting chains is achieved, but also a set of universal quantum gates can be realized with the computational basis formed by the degenerate ground states that are topologically protected against random local perturbations. Physical implementation of our scheme based on a Josephson quantum circuit is presented, where our detailed analysis pinpoints that this scheme is experimentally feasible with the state-of-the-art technology.

We consider topological superconductors and topological insulator/superconductor structures in the presence of multiple static vortices that host Majorana modes and focus on the Majorana tunneling processes between vortices. It is shown that these tunnelings generally lift the degeneracy of the many-body ground state in a non-universal way, sensitive to microscopic details at the smallest length-scales determined by the underlying physical problem. We also discuss an explicit realization of the Jackiw-Rossi zero-mode in a topological insulator/superconductor structure with zero chemical potential. In this case, the exact degeneracy of the many-anyon ground state is protected by an additional chiral symmetry and can be linked to the rigorous index theorem. However, the existence of a non-zero chemical potential, as expected in realistic solid state structures, breaks chiral symmetry and removes protection, which leads to the degeneracy being lifted. Finally, we discuss the implications of our results for the collective states of many-anyon systems. We argue that quantum dynamics of vortices in realistic systems is generally important and may give rise to effective time-dependent gauge factors that enter interaction terms between Majorana modes in many-anyon systems.

A density-matrix analysis is given of some properties of photon echoes arising from nearly degenerate ground and/or excited states. Modulations of the echo intensity as a function of pulse separation time are predicted as well as more complicated effects near level crossings. In particular, the analysis is applied to the 2.06-kG level crossing in the ground state of ruby.

1D topological systems for next-generation electronics

The interaction of the ground states of two closed-shell systems (atom or molecule) is usually repulsive except for a weak long-range van der Waals or electrostatic interaction. But the interaction of an excited state with the ground state of the same fragments can form a strong chemical bond. The bound excited state has been called an excimer or exciplex state and the bound to continuum transition defines an excimer emission system [2.1]. In the case of degenerate open-shell fragments (e.g., P-states, etc.) more than one molecular state will correlate with the degenerate asymptote. Some or all of the curves may be essentially repulsive, but there may also be a bound ground state. The excited state of the open-shell fragment interacting with a ground state fragment can also yield strongly bound molecular states which can be defined as excimer states. The analysis of excimer emission systems in this chapter assumes that the lower repulsive state always arises from ground state fragments which may, however, be degenerate. Rare gas atoms interacting in their ground state provides an example of a single repulsive curve arising from a non-degenerate asymptote. The interaction of two hydrogen atoms in their ground 2 S states will, on the other hand, yield two states a \(^1\Sigma _g^ + \) and a \(^3\Sigma _u^ + \) which are, respectively, strongly bound and repulsive. By generalizing the definition of an excimer emission system to include degenerate asymptotic ground states, we would include almost all possible chemical systems. In order to keep the review within limits, we have restricted consideration to the interactions of either rare gas or Group II atoms with other atomic species for which excimer emission systems have been found and were originally suggested as laser candidates [2.2]. Only two triatomic classes of excimers have been included which are again based on rare gas or Group II molecule fragments.

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with very high accuracy. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. In the crystalline regime (low densities), there exist aperiodic structures that are part of the ground-state manifold but yet are not entropically favored. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit of the canonical ensemble of other potentials with highly degenerate ground states.

We study the spin-one Kitaev model on the honeycomb lattice in the presence of single-ion anisotropies. We consider two types of single ion anisotropies: A ${D}_{111}$ anisotropy which preserves the symmetry between $X, Y$, and $Z$ bonds but violates flux conservation and a ${D}_{100}$ anisotropy that breaks the symmetry between $X, Y$, and $Z$ bonds but preserves flux conservation. We use series expansion methods, degenerate perturbation theory, and exact diagonalization to study these systems. Large positive ${D}_{111}$ anisotropy leads to a simple product ground state with conventional magnonlike excitations, while large negative ${D}_{111}$ leads to a broken symmetry and degenerate ground states. For both signs there is a phase transition at a small $|{D}_{111}|\ensuremath{\approx}0.12$ separating the more conventional phases from the Kitaev spin liquid phase. With large ${D}_{100}$ anisotropy, the ground state is a simple product state, but the model lacks conventional dispersive excitations due to the large number of conservation laws. Large negative ${D}_{100}$ leads to decoupled one-dimensional systems and many degenerate ground states. No evidence of a phase transition is seen in our numerical studies at any finite ${D}_{100}$. Convergence of the series expansion extrapolations all the way to ${D}_{100}=0$ suggests that the nontrivial Kitaev spin liquid is a singular limit of this type of single-ion anisotropy going to zero, which also restores symmetry between the $X, Y$, and $Z$ bonds.

In this paper, we numerically study the non-Abelian statistics of the zero-energy Majorana fermions at the end of Majorana chain and show its application to quantum computing by mapping it to a spin model with special symmetry. In particular, by using transverse-field Ising model with Z2 symmetry, we verify the nontrivial non-Abelian statistics of Majorana fermions. Numerical evidence and comparison in both Majorana representation and spin representation are presented. The degenerate ground states of a symmetry protected spin chain therefore provide a promising platform for topological quantum computation.

Quantum spin liquid is a disordered magnetic state with fractional spin excitations. Its clearest example is found in an exactly solved Kitaev honeycomb model where a spin flip fractionalizes into two types of anyons, quasiparticles that are neither fermions nor bosons: a pair of gauge fluxes and a Majorana fermion. Here we demonstrate this kind of fractionalization in the Kitaev paramagnetic state of the honeycomb magnet $\alpha$-RuCl$_3$. The spin-excitation gap measured by nuclear magnetic resonance consists of the predicted Majorana fermion contribution following the cube of the applied magnetic field, and a finite zero-field contribution matching the predicted size of the gauge-flux gap. The observed fractionalization into gapped anyons survives in a broad range of temperatures and magnetic fields despite inevitable non-Kitaev interactions between the spins, which are predicted to drive the system towards a gapless ground state. The gapped character of both anyons is crucial for their potential application in topological quantum computing.

Topological Quantum Computing: Majorana Fermions and Beyond