397 research outputs found
Abstract Error Groups Via Jones Unitary Braid Group Representations at q=i
In this paper, we classify a type of abstract groups by the central products
of dihedral groups and quaternion groups. We recognize them as abstract error
groups which are often not isomorphic to the Pauli groups in the literature. We
show the corresponding nice error bases equivalent to the Pauli error bases
modulo phase factors. The extension of these abstract groups by the symmetric
group are finite images of the Jones unitary representations (or modulo a phase
factor) of the braid group at q=i or r=4. We hope this work can finally lead to
new families of quantum error correction codes via the representation theory of
the braid group.Comment: 12 page
Classical simulation of Yang-Baxter gates
A unitary operator that satisfies the constant Yang-Baxter equation
immediately yields a unitary representation of the braid group B n for every . If we view such an operator as a quantum-computational gate, then
topological braiding corresponds to a quantum circuit. A basic question is when
such a representation affords universal quantum computation. In this work, we
show how to classically simulate these circuits when the gate in question
belongs to certain families of solutions to the Yang-Baxter equation. These
include all of the qubit (i.e., ) solutions, and some simple families
that include solutions for arbitrary . Our main tool is a
probabilistic classical algorithm for efficient simulation of a more general
class of quantum circuits. This algorithm may be of use outside the present
setting.Comment: 17 pages. Corrected error in proof of Theorem
Anyons in Geometric Models of Matter
We show that the "geometric models of matter" approach proposed by the first
author can be used to construct models of anyon quasiparticles with fractional
quantum numbers, using 4-dimensional edge-cone orbifold geometries with
orbifold singularities along embedded 2-dimensional surfaces. The anyon states
arise through the braid representation of surface braids wrapped around the
orbifold singularities, coming from multisections of the orbifold normal bundle
of the embedded surface. We show that the resulting braid representations can
give rise to a universal quantum computer.Comment: 22 pages LaTe
Braid Matrices and Quantum Gates for Ising Anyons Topological Quantum Computation
We study various aspects of the topological quantum computation scheme based
on the non-Abelian anyons corresponding to fractional quantum hall effect
states at filling fraction 5/2 using the Temperley-Lieb recoupling theory.
Unitary braiding matrices are obtained by a normalization of the degenerate
ground states of a system of anyons, which is equivalent to a modification of
the definition of the 3-vertices in the Temperley-Lieb recoupling theory as
proposed by Kauffman and Lomonaco. With the braid matrices available, we
discuss the problems of encoding of qubit states and construction of quantum
gates from the elementary braiding operation matrices for the Ising anyons
model. In the encoding scheme where 2 qubits are represented by 8 Ising anyons,
we give an alternative proof of the no-entanglement theorem given by Bravyi and
compare it to the case of Fibonacci anyons model. In the encoding scheme where
2 qubits are represented by 6 Ising anyons, we construct a set of quantum gates
which is equivalent to the construction of Georgiev.Comment: 25 pages, 13 figure
Simulation of topological field theories by quantum computers
Quantum computers will work by evolving a high tensor power of a small (e.g.
two) dimensional Hilbert space by local gates, which can be implemented by
applying a local Hamiltonian H for a time t. In contrast to this quantum
engineering, the most abstract reaches of theoretical physics has spawned
topological models having a finite dimensional internal state space with no
natural tensor product structure and in which the evolution of the state is
discrete, H = 0. These are called topological quantum filed theories (TQFTs).
These exotic physical systems are proved to be efficiently simulated on a
quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define
a model of computation stronger than the usual quantum model BQP. 2. TQFTs
provide a radically different way of looking at quantum computation. The rich
mathematical structure of TQFTs might suggest a new quantum algorithm
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
Topological Quantum Computation
The theory of quantum computation can be constructed from the abstract study
of anyonic systems. In mathematical terms, these are unitary topological
modular functors. They underlie the Jones polynomial and arise in
Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in
quantum Hall electron liquids and 2D-magnets are modeled by modular functors,
opening a new possibility for the realization of quantum computers. The chief
advantage of anyonic computation would be physical error correction: An error
rate scaling like e^{-\a\l}, where \l is a length scale, and is
some positive constant. In contrast, the \qpresumptive" qubit-model of
quantum computation, which repairs errors combinatorically, requires a
fantastically low initial error rate (about ) before computation can
be stabilized
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
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