169 research outputs found
On the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm
Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum
adiabatic optimization fails for random instances of NP-complete problems",
arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum
optimization", arXiv:0912.0746] argue that random 4th order perturbative
corrections to the energies of local minima of random instances of NP-complete
problem lead to avoided crossings that cause the failure of quantum adiabatic
algorithm (due to exponentially small gap) close to the end, for very small
transverse field that scales as an inverse power of instance size N. The
theoretical portion of this work does not to take into account the exponential
degeneracy of the ground and excited states at zero field. A corrected analysis
shows that unlike those in the middle of the spectrum, avoided crossings at the
edge would require high [O(1)] transverse fields, at which point the
perturbation theory may become divergent due to quantum phase transition. This
effect manifests itself only in large instances [exp(0.02 N) >> 1], which might
be the reason it had not been observed in the authors' numerical work. While we
dispute the proposed mechanism of failure of quantum adiabatic algorithm, we
cannot draw any conclusions on its ultimate complexity.Comment: 8 pages, 5 figure
Adiabatic Quantum Computing in systems with constant inter-qubit couplings
We propose an approach suitable for solving NP-complete problems via
adiabatic quantum computation with an architecture based on a lattice of
interacting spins (qubits) driven by locally adjustable effective magnetic
fields. Interactions between qubits are assumed constant and
instance-independent, programming is done only by changing local magnetic
fields. Implementations using qubits coupled by magnetic-, electric-dipole and
exchange interactions are discussed.Comment: 10 pages, 10 figures, reference adde
Quantum Adiabatic Evolution Algorithm and Quantum Phase Transition in 3-Satisfiability Problem
In this paper we show that the performance of the quantum adiabatic algorithm
is determined by phase transitions in underlying problem in the presence of
transverse magnetic field . We show that the quantum version of random
Satisfiability problem with 3 bits in a clause (3-SAT) has a first-order
quantum phase transition. We analyze the phase diagram
where is an average number of clauses per binary variable in 3-SAT.
The results are obtained in a closed form assuming replica symmetry and
neglecting time correlations at small values of the transverse field .
In the limit of the value of 5.18 corresponds to
that given by the replica symmetric treatment of a classical random 3-SAT
problem. We demonstrate the qualitative similarity between classical and
quantum versions of this problem.Comment: 30 pages, 7 figure
Complete spin polarization of degenerate electrons in semiconductors near ferromagnetic contacts
We show that spin polarization of electron density in nonmagnetic degenerate
semiconductors can achieve 100%. This effect is realized in
ferromagnet-semiconductor - junctions even at moderate spin
selectivity of the contact when the electrons are extracted from the
heavily doped semiconductor into the ferromagnet. We derived a general
equation relating spin polarization of the current to that of the electron
density in nonmagnetic semiconductors. We found that the effect of the complete
spin polarization is achieved near - interface when an effective
diffusion coefficient goes to zero in this region while the diffusion current
remains finite
Complete spin extraction from semiconductors near ferromagnet-semiconductor interfaces
We show that spin polarization of electrons in nonmagnetic semiconductors
near specially tailored ferromagnet-semiconductor junctions can achieve 100%.
This effect is realized even at moderate spin injection coefficients of the
contact when these coefficients only weakly depend on the current. The effect
of complete spin extraction occurs at relatively strong electric fields and
arises from a reduction of spin penetration length due to the drift of
electrons from a semiconductor towards the spin-selective tunnel junction
Electronic Control and Readout of Qubit States in Solid State Quantum Computing Systems
We demonstrate that an junction is the most suitable candidate
for electronic control and readout of qubit states in quantum computing systems
based on shallow impurities. The signature of this system is that the
regions serve as metallic electrodes separated form the region by a
self-induced barrier (internal workfunction). The system mimics the
properties of a metal-vacuum-metal junction with the qubit (impurity atom)
placed in a ``vacuum'' -region between two ``metallic'' electrodes. We
will show that the self-induced barrier exists in a sufficiently wide range of
the concentration of dopants in the -semiconductor (e.g. up to
cm for Si) and its height can be controlled by tuning the doping level.
A shallow donor placed in a vacuum -region will be populated with one
electron in equilibrium. In the case of Li donor in Si the -electrodes
will be used for a precision placement of the Li atom during the growth
process; for voltage control and manipulation of the qubit states; and for a
qubit readout by means of the optically stimulated resonant tunnelling. Another
important feature of our system is that the qubit states (first two lowest
energy levels of Li in Si) are separated by an energy gap from a continuum of
the many-body states of the controlling electrodes
Simulations of the adiabatic quantum optimization for the Set Partition Problem
We analyze the complexity of the quantum optimization algorithm based on
adiabatic evolution for the set partition problem. We introduce a cost function
defined on a logarithmic scale of the partition residues so that the total
number of values of the cost function is of the order of the problem size. We
simulate the behavior of the algorithm by numerical solution of the
time-dependent Schroedinger equation as well as the stationary equation for the
adiabatic eigenvalues. The numerical results for the time-dependent quantum
evolution indicate that the complexity of the algorithm scales exponentially
with the problem size.This result appears to contradict the recent numerical
results for complexity of quantum adiabatic algorithm applied to a different
NP-complete problem (Farhi et al, Science 292, p.472 (2001)).Comment: 13 pages, 5 figures Added simulation results for n=17 bit
Identification of nonlinear noisy dynamics of an ecosystem from observations of one of its trajectory components
The problem of determining dynamical models and trajectories that describe
observed time-series data allowing for the understanding, prediction and
possibly control of complex systems in nature is of a great interest in a wide
variety of fields. Often, however, only part of the system's dynamical
variables can be measured, the measurements are corrupted by noise and the
dynamics is complicated by an interplay of nonlinearity and random
perturbations. The problem of dynamical inference in these general settings is
challenging researchers for decades. We solve this problem by applying a
path-integral approach to fluctuational dynamics, and show that, given the
measurements, the system trajectory can be obtained from the solution of the
certain auxiliary Hamiltonian problem in which measured data act effectively as
a control force driving the estimated trajectory toward the most probable one
that provides a minimum to certain mechanical action. The dependance of the
minimum action on the model parameters determines the statistical distribution
in the model space consistent with the measurements. We illustrate the
efficiency of the approach by solving an intensively studied problem from the
population dynamics of predator-prey system where the prey populations may be
observed while the predator populations or even their number is difficult or
impossible to estimate. We apply our approach to recover both the unknown
dynamics of predators and model parameters (including parameters that are
traditionally very difficult to estimate) directly from measurements of the
prey dynamics. We provide a comparison of our method with the Markov Chain
Monte Carlo technique.Comment: 30 pages, 7 figure
Approximating satisfiability transition by suppressing fluctuations
Using methods and ideas from statistical mechanics, we propose a simple
method for obtaining rigorous upper bounds for satisfiability transition in
random boolean expressions composed of N variables and M clauses with K
variables per clause. Determining the location of satisfiability threshold
for a number of difficult combinatorial problems is a major open
problem in the theory of random graphs. The method is based on identification
of the core -- a subexpression (subgraph) that has the same satisfiability
properties as the original expression. We formulate self-consistency equations
that determine macroscopic parameters of the core and compute an improved
annealing bound. We illustrate the method for three sample problems: K-XOR-SAT,
K-SAT and positive 1-in-K-SAT.Comment: 31 pages, 6 figure
Size dependence of the minimum excitation gap in the Quantum Adiabatic Algorithm
We study the typical (median) value of the minimum gap in the quantum version
of the Exact Cover problem using Quantum Monte Carlo simulations, in order to
understand the complexity of the quantum adiabatic algorithm (QAA) for much
larger sizes than before. For a range of sizes, N <= 128, where the classical
Davis-Putnam algorithm shows exponential median complexity, the QAA shows
polynomial median complexity. The bottleneck of the algorithm is an isolated
avoided crossing point of a Landau-Zener type (collision between the two lowest
energy levels only).Comment: 4 pages, 5 figure
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