829 research outputs found
An Isomonodromy Cluster of Two Regular Singularities
We consider a linear matrix ODE with two coalescing regular
singularities. This coalescence is restricted with an isomonodromy condition
with respect to the distance between the merging singularities in a way
consistent with the ODE. In particular, a zero-distance limit for the ODE
exists. The monodromy group of the limiting ODE is calculated in terms of the
original one. This coalescing process generates a limit for the corresponding
nonlinear systems of isomonodromy deformations. In our main example the latter
limit reads as , where is the -th Painlev\'e equation. We
also discuss some general problems which arise while studying the
above-mentioned limits for the Painlev\'e equations.Comment: 44 pages, 8 figure
Experimental Quantum Process Discrimination
Discrimination between unknown processes chosen from a finite set is
experimentally shown to be possible even in the case of non-orthogonal
processes. We demonstrate unambiguous deterministic quantum process
discrimination (QPD) of non-orthogonal processes using properties of
entanglement, additional known unitaries, or higher dimensional systems. Single
qubit measurement and unitary processes and multipartite unitaries (where the
unitary acts non-separably across two distant locations) acting on photons are
discriminated with a confidence of in all cases.Comment: 4 pages, 3 figures, comments welcome. Revised version includes
multi-partite QP
Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients
We consider deformations of and matrix linear ODEs with
rational coefficients with respect to singular points of Fuchsian type which
don't satisfy the well-known system of Schlesinger equations (or its natural
generalization). Some general statements concerning reducibility of such
deformations for ODEs are proved. An explicit example of the general
non-Schlesinger deformation of -matrix ODE of the Fuchsian type with
4 singular points is constructed and application of such deformations to the
construction of special solutions of the corresponding Schlesinger systems is
discussed. Some examples of isomonodromy and non-isomonodromy deformations of
matrix ODEs are considered. The latter arise as the compatibility
conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.
Universal 2-local Hamiltonian Quantum Computing
We present a Hamiltonian quantum computation scheme universal for quantum
computation (BQP). Our Hamiltonian is a sum of a polynomial number (in the
number of gates L in the quantum circuit) of time-independent, constant-norm,
2-local qubit-qubit interaction terms. Furthermore, each qubit in the system
interacts only with a constant number of other qubits. The computer runs in
three steps - starts in a simple initial product-state, evolves it for time of
order L^2 (up to logarithmic factors) and wraps up with a two-qubit
measurement. Our model differs from the previous universal 2-local Hamiltonian
constructions in that it does not use perturbation gadgets, does not need large
energy penalties in the Hamiltonian and does not need to run slowly to ensure
adiabatic evolution.Comment: recomputed the necessary number of interactions, new geometric
layout, added reference
Quantum speedup of classical mixing processes
Most approximation algorithms for #P-complete problems (e.g., evaluating the
permanent of a matrix or the volume of a polytope) work by reduction to the
problem of approximate sampling from a distribution over a large set
. This problem is solved using the {\em Markov chain Monte Carlo} method: a
sparse, reversible Markov chain on with stationary distribution
is run to near equilibrium. The running time of this random walk algorithm, the
so-called {\em mixing time} of , is as shown
by Aldous, where is the spectral gap of and is the minimum
value of . A natural question is whether a speedup of this classical
method to , the diameter of the graph
underlying , is possible using {\em quantum walks}.
We provide evidence for this possibility using quantum walks that {\em
decohere} under repeated randomized measurements. We show: (a) decoherent
quantum walks always mix, just like their classical counterparts, (b) the
mixing time is a robust quantity, essentially invariant under any smooth form
of decoherence, and (c) the mixing time of the decoherent quantum walk on a
periodic lattice is , which is indeed
and is asymptotically no worse than the
diameter of (the obvious lower bound) up to at most a logarithmic
factor.Comment: 13 pages; v2 revised several part
Modulated Floquet Topological Insulators
Floquet topological insulators are topological phases of matter generated by
the application of time-periodic perturbations on otherwise conventional
insulators. We demonstrate that spatial variations in the time-periodic
potential lead to localized quasi-stationary states in two-dimensional systems.
These states include one-dimensional interface modes at the nodes of the
external potential, and fractionalized excitations at vortices of the external
potential. We also propose a setup by which light can induce currents in these
systems. We explain these results by showing a close analogy to px+ipy
superconductors
Topological Quantum Computing with Only One Mobile Quasiparticle
In a topological quantum computer, universal quantum computation is performed
by dragging quasiparticle excitations of certain two dimensional systems around
each other to form braids of their world lines in 2+1 dimensional space-time.
In this paper we show that any such quantum computation that can be done by
braiding identical quasiparticles can also be done by moving a single
quasiparticle around n-1 other identical quasiparticles whose positions remain
fixed.Comment: 4 pages, 5 figure
Topological Quantum Compiling
A method for compiling quantum algorithms into specific braiding patterns for
non-Abelian quasiparticles described by the so-called Fibonacci anyon model is
developed. The method is based on the observation that a universal set of
quantum gates acting on qubits encoded using triplets of these quasiparticles
can be built entirely out of three-stranded braids (three-braids). These
three-braids can then be efficiently compiled and improved to any required
accuracy using the Solovay-Kitaev algorithm.Comment: 20 pages, 20 figures, published versio
Braid Topologies for Quantum Computation
In topological quantum computation, quantum information is stored in states
which are intrinsically protected from decoherence, and quantum gates are
carried out by dragging particle-like excitations (quasiparticles) around one
another in two space dimensions. The resulting quasiparticle trajectories
define world-lines in three dimensional space-time, and the corresponding
quantum gates depend only on the topology of the braids formed by these
world-lines. We show how to find braids that yield a universal set of quantum
gates for qubits encoded using a specific kind of quasiparticle which is
particularly promising for experimental realization.Comment: 4 pages, 4 figures, minor revision
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