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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 28 (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.
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 15 (1 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].
Periodic table for topological insulators and superconductors
, 901
"... Abstract. 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 re ..."
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Cited by 4 (0 self)
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Abstract. 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. Keywords: Topological phase, Ktheory, Khomology, Clifford algebra, Bott periodicity
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 3 (1 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
doi:10.1006/aphy.2002.6254 Fermionic Quantum Computation
, 2002
"... We define a model of quantum computation with local fermionic modes (LFMs)—sites which can be either empty or occupied by a fermion. With the standard correspondence between the Foch space of m LFMs and the Hilbert space of m qubits, simulation of one fermionic gate takes O(m) qubit gates and vice v ..."
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We define a model of quantum computation with local fermionic modes (LFMs)—sites which can be either empty or occupied by a fermion. With the standard correspondence between the Foch space of m LFMs and the Hilbert space of m qubits, simulation of one fermionic gate takes O(m) qubit gates and vice versa. We show that using different encodings, the simulation cost can be reduced to O(log m) and a constant, respectively. Nearest neighbors fermionic gates on a graph of bounded degree can be simulated at a constant cost. A universal set of fermionic gates is found. We also study computation with Majorana fermions which are basically halves of LFMs. Some connection to qubit quantum codes is made. C ○ 2002 Elsevier Science (USA)
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