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35
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 85 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
Periodic table for topological insulators and superconductors
, 2009
"... Gapped phases of noninteracting fermions, with and without charge conservation and timereversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Cliffo ..."
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Cited by 57 (0 self)
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Gapped phases of noninteracting fermions, with and without charge conservation and timereversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of Khomology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the Ktheoretic classification is stable to interactions, but a counterexample is also given.
Classical simulation of noninteractingfermion quantum circuits
 Phys. Rev. A
"... We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [1] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend ..."
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Cited by 26 (2 self)
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We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant [1] corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend the result to noninteracting fermions with arbitrary pairwise interactions, where gates can be conditioned on outcomes of complete von Neumann measurements in the computational basis on other fermionic modes in the circuit. This last result is in remarkable contrast with the case of noninteracting bosons where universal quantum computation can be achieved by allowing gates to be conditioned on classical bits [2].
Search for majorana fermions in superconductors
 arXiv:1112.1950
, 2011
"... Majorana fermions (particles which are their own antiparticle) may or may not exist in Nature as elementary building blocks, but in condensed matter they can be constructed out of electron and hole excitations. What is needed is a superconductor to hide the charge difference, and a topological (Berr ..."
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Cited by 8 (0 self)
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Majorana fermions (particles which are their own antiparticle) may or may not exist in Nature as elementary building blocks, but in condensed matter they can be constructed out of electron and hole excitations. What is needed is a superconductor to hide the charge difference, and a topological (Berry) phase to eliminate the energy difference from zeropoint motion. A pair of widely separated Majorana fermions, bound to magnetic or electrostatic defects, has nonAbelian exchange statistics. A qubit encoded in this Majorana pair is expected to have an unusually long coherence time. We discuss strategies to detect Majorana fermions in a topological superconductor, as well as possible applications in a quantum computer. The status of the experimental search is reviewed. scheduled for vol. 4 (2013) of Annual Review of Condensed Matter Physics Contents I. What are they? 1 A. Their origin in particle physics 1 B. Their emergence in superconductors 1 C. Their potential for quantum computing 2
Teleportation by a Majorana Medium
, 2008
"... It is argued that Majorana zero modes in a system of quantum fermions can mediate a teleportationlike process with the actual transfer of electronic material between wellseparated points. The problem is formulated in the context of a quasirealistic and exactly solvable model of a quantum wire embe ..."
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Cited by 4 (2 self)
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It is argued that Majorana zero modes in a system of quantum fermions can mediate a teleportationlike process with the actual transfer of electronic material between wellseparated points. The problem is formulated in the context of a quasirealistic and exactly solvable model of a quantum wire embedded in a bulk pwave superconductor. Teleportation by quantum tunneling 1 in one form or another has been the physicist’s dream since the invention of the quantum theory. The simplest idea makes use of the fact that the quantum wavefunction can have support in classically forbidden regions and can thus reach across apparent barriers. Wherever the wavefunction has support, the object whose probability amplitude it describes can in principle be found. Of course, the typical profile of a wavefunction inside a forbidden region decays exponentially with distance, so its amplitude on the other side of that region should be vanishingly small, particularly if any appreciable distance is involved. A slightly
Topological phases and quantum computation
, 904
"... 2 Topological phenomena in 1D: boundary modes in the Majorana chain 3 2.1 Nature of topological degeneracy (spin language) 4 ..."
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2 Topological phenomena in 1D: boundary modes in the Majorana chain 3 2.1 Nature of topological degeneracy (spin language) 4
Engineered open systems and quantum simulations with atoms and ions
 Advances In Atomic, Molecular, and Optical Physics
, 2012
"... The enormous experimental progress in atomic, molecular and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even manybody ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthin ..."
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The enormous experimental progress in atomic, molecular and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even manybody ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthinkable some years ago. This versatile set of tools has enabled the development of the wellestablished concept of engineering of manybody Hamiltonians in various physical platforms. These available tools, however, can also be harnessed to extend the scenario of Hamiltonian engineering to a more general Liouvillian setting, which in addition to coherent dynamics also includes controlled dissipation in manybody quantum systems. Here, we review recent theoretical and experimental progress in different directions along these lines, with a particular focus on physical realizations with systems
Disorderassisted error correction in Majorana chains
 Comm. Math.Phys
"... It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions the simplest toy model of a quantum m ..."
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It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions the simplest toy model of a quantum memory. Disorder takes the form of a random sitedependent chemical potential. The corresponding oneparticle problem is a onedimensional Anderson model with disorder in the hopping amplitudes. We focus on the zerotemperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an errorcorrection step. Assuming dynamical localization of the oneparticle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the oneparticle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.