3,911 research outputs found
Simulation of anyons with tensor network algorithms
Interacting systems of anyons pose a unique challenge to condensed matter
simulations due to their non-trivial exchange statistics. These systems are of
great interest as they have the potential for robust universal quantum
computation, but numerical tools for studying them are as yet limited. We show
how existing tensor network algorithms may be adapted for use with systems of
anyons, and demonstrate this process for the 1-D Multi-scale Entanglement
Renormalisation Ansatz (MERA). We apply the MERA to infinite chains of
interacting Fibonacci anyons, computing their scaling dimensions and local
scaling operators. The scaling dimensions obtained are seen to be in agreement
with conformal field theory. The techniques developed are applicable to any
tensor network algorithm, and the ability to adapt these ansaetze for use on
anyonic systems opens the door for numerical simulation of large systems of
free and interacting anyons in one and two dimensions.Comment: Fixed typos, matches published version. 16 pages, 21 figures, 4
tables, RevTeX 4-1. For a related work, see arXiv:1006.247
Braiding and entanglement in spin networks: a combinatorial approach to topological phases
The spin network quantum simulator relies on the su(2) representation ring
(or its q-deformed counterpart at q= root of unity) and its basic features
naturally include (multipartite) entanglement and braiding. In particular,
q-deformed spin network automata are able to perform efficiently approximate
calculations of topological invarians of knots and 3-manifolds. The same
algebraic background is shared by 2D lattice models supporting topological
phases of matter that have recently gained much interest in condensed matter
physics. These developments are motivated by the possibility to store quantum
information fault-tolerantly in a physical system supporting fractional
statistics since a part of the associated Hilbert space is insensitive to local
perturbations. Most of currently addressed approaches are framed within a
'double' quantum Chern-Simons field theory, whose quantum amplitudes represent
evolution histories of local lattice degrees of freedom.
We propose here a novel combinatorial approach based on `state sum' models of
the Turaev-Viro type associated with SU(2)_q-colored triangulations of the
ambient 3-manifolds. We argue that boundary 2D lattice models (as well as
observables in the form of colored graphs satisfying braiding relations) could
be consistently addressed. This is supported by the proof that the Hamiltonian
of the Levin-Wen condensed string net model in a surface Sigma coincides with
the corresponding Turaev-Viro amplitude on Sigma x [0,1] presented in the last
section.Comment: Contributed to Quantum 2008: IV workshop ad memoriam of Carlo Novero
19-23 May 2008 - Turin, Ital
Simulation of braiding anyons using Matrix Product States
Anyons exist as point like particles in two dimensions and carry braid
statistics which enable interactions that are independent of the distance
between the particles. Except for a relatively few number of models which are
analytically tractable, much of the physics of anyons remain still unexplored.
In this paper, we show how U(1)-symmetry can be combined with the previously
proposed anyonic Matrix Product States to simulate ground states and dynamics
of anyonic systems on a lattice at any rational particle number density. We
provide proof of principle by studying itinerant anyons on a one dimensional
chain where no natural notion of braiding arises and also on a two-leg ladder
where the anyons hop between sites and possibly braid. We compare the result of
the ground state energies of Fibonacci anyons against hardcore bosons and
spinless fermions. In addition, we report the entanglement entropies of the
ground states of interacting Fibonacci anyons on a fully filled two-leg ladder
at different interaction strength, identifying gapped or gapless points in the
parameter space. As an outlook, our approach can also prove useful in studying
the time dynamics of a finite number of nonabelian anyons on a finite
two-dimensional lattice.Comment: Revised version: 20 pages, 14 captioned figures, 2 new tables. We
have moved a significant amount of material concerning symmetric tensors for
anyons --- which can be found in prior works --- to Appendices in order to
streamline our exposition of the modified Anyonic-U(1) ansat
Combining Topological Hardware and Topological Software: Color Code Quantum Computing with Topological Superconductor Networks
We present a scalable architecture for fault-tolerant topological quantum
computation using networks of voltage-controlled Majorana Cooper pair boxes,
and topological color codes for error correction. Color codes have a set of
transversal gates which coincides with the set of topologically protected gates
in Majorana-based systems, namely the Clifford gates. In this way, we establish
color codes as providing a natural setting in which advantages offered by
topological hardware can be combined with those arising from topological
error-correcting software for full-fledged fault-tolerant quantum computing. We
provide a complete description of our architecture including the underlying
physical ingredients. We start by showing that in topological superconductor
networks, hexagonal cells can be employed to serve as physical qubits for
universal quantum computation, and present protocols for realizing
topologically protected Clifford gates. These hexagonal cell qubits allow for a
direct implementation of open-boundary color codes with ancilla-free syndrome
readout and logical -gates via magic state distillation. For concreteness,
we describe how the necessary operations can be implemented using networks of
Majorana Cooper pair boxes, and give a feasibility estimate for error
correction in this architecture. Our approach is motivated by nanowire-based
networks of topological superconductors, but could also be realized in
alternative settings such as quantum Hall-superconductor hybrids.Comment: 24 pages, 24 figure
Universal quantum computation with ordered spin-chain networks
It is shown that anisotropic spin chains with gapped bulk excitations and
magnetically ordered ground states offer a promising platform for quantum
computation, which bridges the conventional single-spin-based qubit concept
with recently developed topological Majorana-based proposals. We show how to
realize the single-qubit Hadamard, phase, and pi/8 gates as well as the
two-qubit CNOT gate, which together form a fault-tolerant universal set of
quantum gates. The gates are implemented by judiciously controlling Ising
exchange and magnetic fields along a network of spin chains, with each
individual qubit furnished by a spin-chain segment. A subset of single-qubit
operations is geometric in nature, relying on control of anisotropy of spin
interactions rather than their strength. We contrast topological aspects of the
anisotropic spin-chain networks to those of p-wave superconducting wires
discussed in the literature.Comment: 9 pages, 3 figure
Conserved Quantities in Background Independent Theories
We discuss the difficulties that background independent theories based on
quantum geometry encounter in deriving general relativity as the low energy
limit. We follow a geometrogenesis scenario of a phase transition from a
pre-geometric theory to a geometric phase which suggests that a first step
towards the low energy limit is searching for the effective collective
excitations that will characterize it. Using the correspondence between the
pre-geometric background independent theory and a quantum information
processor, we are able to use the method of noiseless subsystems to extract
such coherent collective excitations. We illustrate this in the case of locally
evolving graphs.Comment: 11 pages, 3 figure
Jordan-Wigner transformations for tree structures
The celebrated Jordan--Wigner transformation provides an efficient mapping
between spin chains and fermionic systems in one dimension. Here we extend this
spin-fermion mapping to arbitrary tree structures, which enables mapping
between fermionic and spin systems with nearest-neighbor coupling. The mapping
is achieved with the help of additional spins at the junctions between
one-dimensional chains. This property allows for straightforward simulation of
Majorana braiding in spin or qubit systems
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