Jason Alicea

Non-Abelian statistics and topological quantum information processing in 1D wire networks

The synthesis of a quantum computer remains an ongoing challenge in modern physics. Whereas decoherence stymies most approaches, topological quantum computation schemes evade decoherence at the hardware level by storing quantum information non-locally. Here we establish that a key operation—braiding of non-Abelian anyons—can be implemented using one-dimensional semiconducting wires. Such wires can be driven into a topological phase supporting long-sought particles known as Majorana fermions that can encode topological qubits. We show that in wire networks, Majorana fermions can be meaningfully braided by simply adjusting gate voltages, and that they exhibit non-Abelian statistics like vortices in a p+ip superconductor. We propose experimental set-ups that enable probing of the Majorana fusion rules and the efficient exchange of arbitrary numbers of Majorana fermions. This work should open a new direction in topological quantum computation that benefits from physical transparency and experimental feasibility.

Majorana fermions in a tunable semiconductor device

The experimental realization of Majorana fermions presents an important problem due to their non-Abelian nature and potential exploitation for topological quantum computation. Very recently Sau et al. [Phys. Rev. Lett. 104, 040502 (2010)] demonstrated that a topological superconducting phase supporting Majorana fermions can be realized using surprisingly conventional building blocks: a semiconductor quantum well coupled to an s-wave superconductor and a ferromagnetic insulator. Here we propose an alternative setup, wherein a topological superconducting phase is driven by applying an in-plane magnetic field to a (110)-grown semiconductor coupled only to an s-wave superconductor. This device offers a number of advantages, notably a simpler architecture and the ability to tune across a quantum phase transition into the topological superconducting state while still largely avoiding unwanted orbital effects. Experimental feasibility of both setups is discussed in some detail.

Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes

Starting from the graphene lattice tight-binding Hamiltonian with an on-site U and long-range Coulomb repulsion, we derive an interacting continuum Dirac theory governing the low-energy behavior of graphene in an applied magnetic field. Initially, we consider a clean graphene system within this effective theory and explore integer quantum Hall ferromagnetism stabilized by exchange from the long-range Coulomb repulsion. We study in detail the ground state and excitations at = 0 and = ± 1, taking into account small symmetrybreaking terms that arise from the lattice-scale interactions, and also explore the ground states selected at = ± 3, ±4, and ±5. We argue that the ferromagnetic regime may not yet be realized in current experimental samples, which at the above filling factors perhaps remain paramagnetic due to strong disorder. In an attempt to access the latter regime where the role of exchange is strongly suppressed by disorder, we apply Hartree theory to study the effects of interactions. Here, we find that Zeeman splitting together with symmetry-breaking interactions can in principle produce integer quantum Hall states in a paramagnetic system at = 0, ±1, and ±4, but not at = ± 3 or ±5, consistent with recent experiments in high magnetic fields. We make predictions for the activation energies in these quantum Hall states which will be useful for determining their true origin.

Exotic non-Abelian anyons from conventional fractional quantum Hall states

Non-abelian anyons--particles whose exchange noncommutatively transforms a systems quantum state--are widely sought for the exotic fundamental physics they harbour and for quantum computing applications. Numerous blueprints now exist for stabilizing the simplest type of non-anyon, defects binding Majorana modes, by interfacing widely available materials. Here we introduce a device fabricated from conventional fractional quantum Hall states and s-wave superconductors that supports exotic non-defects binding parafermionic zero modes, which generalize Majorana bound states. We show that these new modes can be experimentally identified (and distinguished from Majoranas) using Josephson measurements. We also provide a practical recipe for braiding parafermionic zero modes and show that they give rise to non-statistics. Interestingly, braiding in our setup produces a richer set of topologically protected operations when compared with the Majorana case. As a byproduct, we establish a new, experimentally realistic Majorana platform in weakly spin-orbit-coupled materials such as gallium arsenide.

Milestones toward Majorana-based quantum computing

We introduce a scheme for preparation, manipulation, and readout of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate-control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current; (2) validation of a prototype topological qubit; and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the systems excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and readout schemes as well.

Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes

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 of the topological superconducting wires since they are all parallel; (2) topological T junctions are not used, obviating possible difficulties in their fabrication and utilization; (3) quasiparticle poisoning is abated by the charging energy; (4) Clifford operations are executed by a relatively standard measurement: detection of corrections to quantum dot energy, charge, or differential capacitance induced by quantum fluctuations; (5) it is compatible with strategies for producing good approximate magic states.

Order-by-disorder and spiral spin-liquid in frustrated diamond-lattice antiferromagnets

Frustration refers to competition between different interactions that cannot be simultaneously satisfied—a familiar feature in many magnetic solids. Strong frustration leads to highly degenerate ground states and a large suppression of ordering by fluctuations. Key challenges in frustrated magnetism include the characterization of the fluctuating spin-liquid regime and determination of the mechanism of eventual order at lower temperature. Here, we study a model of a diamond-lattice antiferromagnet appropriate for numerous spinel materials. With sufficiently strong frustration, a massive ground-state degeneracy develops amongst spirals whose propagation wavevectors reside on a continuous two-dimensional ‘spiral surface’ in momentum space. We argue that an important ordering mechanism is entropic splitting of the degenerate ground states, an elusive phenomenon called ‘order by disorder’. A broad spiral spin-liquid regime emerges at higher temperatures, where the underlying spiral surface can be directly revealed through spin correlations. We discuss the agreement between these predictions and the well-characterized spinel MnSc2S4.

Universal topological quantum computation from a superconductor-abelian quantum hall heterostructure

Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Greens observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that---unlike Ising anyons---allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kanes construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a two-fold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types yet depending on non-universal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.

Interaction effects in topological superconducting wires supporting Majorana fermions

Among the broad spectrum of systems predicted to exhibit topological superconductivity and Majorana fermions, one-dimensional wires with strong spin-orbit coupling provide one of the most promising experimental candidates. Here we investigate the fate of the topological superconducting phase in such wires when repulsive interactions are present. Using a combination of density matrix renormalization group, bosonization, and Hartree\char21{}Fock techniques, we demonstrate that while interactions degrade the bulk gap\char22{}consistent with recent results of Gangadharaiah et al.\char22{}they also greatly expand the parameter range over which the topological phase arises. In particular, we show that with interactions this phase can be accessed over a broader chemical potential window, thereby leading to greater immunity against disorder-induced chemical potential fluctuations in the wire. We also suggest that in certain wires strong interactions may allow Majorana fermions to be generated without requiring a magnetic field.

Adiabatic manipulations of Majorana fermions in a three-dimensional network of quantum wires

It has been proposed that localized zero-energy Majorana states can be realized in a two-dimensional network of quasi-one-dimensional semiconductor wires that are proximity coupled to a bulk superconductor. The wires should have strong spin-orbit coupling with appropriate symmetry, and their electrons should be partially polarized by a strong Zeeman field. Then, if the Fermi level is in an appropriate range, the wire can be in a topological superconducting phase, with Majorana states that occur at wire ends and at Y junctions, where three topological superconductor segments may be joined. Here we generalize these ideas to consider a three-dimensional network. The positions of Majorana states can be manipulated, and their non-Abelian properties made visible, by using external gates to selectively deplete portions of the network or by physically connecting and redividing wire segments. Majorana states can also be manipulated by reorientations of the Zeeman field on a wire segment, by physically rotating the wire about almost any axis, or by evolution of the phase of the order parameter in the proximity-coupled superconductor. We show how to keep track of sign changes in the zero-energy Hilbert space during adiabatic manipulations by monitoring the evolution of each Majorana state separately, rather than keeping track of the braiding of all possible pairs. This has conceptual advantages in the case of a three-dimensional network, and may be computationally useful even in two dimensions, if large numbers of Majorana sites are involved.