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 that can perform the initialization-braiding-measurement steps required to demonstrate non-Abelian statistics. We introduce the Random Access Majorana Memory, a scalable circuit that can perform a joint parity measurement on Majoranas belonging to a selection of topological qubits. Such multi-qubit measurements allow for the efficient creation of highly entangled states and simplify quantum error correction protocols by avoiding the need for ancilla qubits.After the first signatures were reported [1][2][3][4] of Majorana bound states in superconducting nanowires [5][6][7], the quest for non-Abelian braiding statistics [8][9][10][11] has intensified. Much interest towards Majorana fermions arises from their technological potential in fault-tolerant quantum computation [12][13][14][15][16]. Their non-Abelian exchange statistics would allow to perform quantum gates belonging to the Clifford group with extremely good accuracy. Moreover, topological qubits encoded non-locally in well-separated Majorana bound states would be resilient against many sources of decoherence. Even without the applications in quantum information processing, observing a new type of quantum statistics would be a milestone in the history of physics.The two central issues for the application of Majorana fermions are (i) how to unambiguously demonstrate their non-Abelian exchange statistics and (ii) how to exploit their full potential for quantum information processing. The first issue requires an elementary circuit that can perform three tasks: initialization of a qubit, braiding (exchange) of two Majoranas, and finally measurement (readout) of the qubit. In view of the second issue, this circuit should be scalable and serve as a first step towards universal fault-tolerant quantum computation.Here we present such a circuit, using a superconducting charge qubit in a transmission line resonator (transmon [17][18][19][20]) to initialize, control, and measure the topological qubit. In such a hybrid system, named top-transmon [21], the long-range Coulomb couplings of Majorana fermions can be used to braid them and to read out their fermion parity [21,22]. While there exist several proposals to control or measure Majorana fermions in nanowires [11,[21][22][23][24][25][26][27][28][29][30][31][32], combining braiding and measurement without local adjustment of microscopic parameters remains a challenge. We show that full macroscopic control is possible if during the...
We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are charge-neutral quasiparticles (equal to their own antiparticle), they have an effective long-range interaction through the even-odd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the non-Abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes.
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multi-component Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases, based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete D 3 gauge group, are presented.
Tensor networks, and in particular Projected Entangled Pair States (PEPS), are a powerful tool for the study of quantum many body physics, thanks to both their built-in ability of classifying and studying symmetries, and the efficient numerical calculations they allow. In this work, we introduce a way to extend the set of symmetric PEPS in order to include local gauge invariance and investigate lattice gauge theories with fermionic matter. To this purpose, we provide as a case study and first example, the construction of a fermionic PEPS, based on Gaussian schemes, invariant under both global and local U (1) gauge transformations. The obtained states correspond to a truncated U (1) lattice gauge theory in 2 + 1 dimensions, involving both the gauge field and fermionic matter. For the global symmetry (pure fermionic) case, these PEPS can be studied in terms of spinless fermions subject to a p-wave superconducting pairing. For the local symmetry (fermions and gauge fields) case, we find confined and deconfined phases in the pure gauge limit, and we discuss the screening properties of the phases arising in the presence of dynamical matter.arXiv:1507.08837v2 [quant-ph]
We show that non-abelian potentials acting on ultracold gases with two hyperfine levels can give rise to ground states with non-abelian excitations. We consider a realistic gauge potential for which the Landau levels can be exactly determined: the non-abelian part of the vector potential makes the Landau levels non-degenerate. In the presence of strong repulsive interactions, deformed Laughlin ground states occur in general. However, at the degeneracy points of the Landau levels, non-abelian quantum Hall states appear: these ground states, including deformed Moore-Read states (characterized by Ising anyons as quasi-holes), are studied for both fermionic and bosonic gases.The quest for states of matter having non-abelian excitations has been very intense in the past two decades, motivated by interest for the topological properties of such correlated states [1,2] and by their relevance for the implementation of topological quantum computation schemes [2].Quantum Hall systems provide a major arena in which investigating non-abelian anyons, and a huge amount of work has been devoted to characterize which quantum Hall states are non-abelian and to propose (mainly interferometric) experiments to test the non-abelian nature of such states A significant and complementary issue consists in finding other experimental systems suitable to simulate the quantum Hall physics and realize non-abelian anyons. In this respect, ultracold atomic systems provide a natural candidate [5,6], due to the possibility of using them as simulators of many-body systems [7]. Two essential (but in general not sufficient) ingredients are available in ultracold atomic systems. First, one can simulate artificial magnetic fields by using
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