2,400 research outputs found
Dynamical formation and manipulation of Majorana fermions in driven quantum wires
Controlling the dynamics of Majorana fermions (MF) subject to time-varying
driving fields is of fundamental importance for the practical realization of
topological quantum computing. In this work we study how it is possible to
dynamically generate and maintain the topological phase in one-dimensional
superconducting nanowires after the temporal variation of the Hamiltonian
parameters. Remarkably we show that for a sudden quench the system can never
relax towards a state exhibiting fully developed MF, independently of the
initial and final Hamiltonians. Only for sufficiently slow protocols the system
behaves adiabatically, and the topological phase can be reached. Finally we
address the crucial question of how "adiabatic" a protocol must be in order to
manipulate the MF inside the topological phase without deteriorating their
Majorana character.Comment: 5 pages, 4 eps figure
Adiabatic approximation in the second quantized formulation
Recently there have been some controversies about the criterion of the
adiabatic approximation. It is shown that an approximate diagonalization of the
effective Hamiltonian in the second quantized formulation gives rise to a
reliable and unambiguous criterion of the adiabatic approximation. This is
illustrated for the model of Marzlin and Sanders and a model related to the
geometric phase which can be exactly diagonalized in the present sense.Comment: 16 page
Adiabatic Quantum Computation with a 1D projector Hamiltonian
Adiabatic quantum computation is based on the adiabatic evolution of quantum
systems. We analyse a particular class of qauntum adiabatic evolutions where
either the initial or final Hamiltonian is a one-dimensional projector
Hamiltonian on the corresponding ground state. The minimum energy gap which
governs the time required for a successful evolution is shown to be
proportional to the overlap of the ground states of the initial and final
Hamiltonians. We show that such evolutions exhibit a rapid crossover as the
ground state changes abruptly near the transition point where the energy gap is
minimum. Furthermore, a faster evolution can be obtained by performing a
partial adiabatic evolution within a narrow interval around the transition
point. These results generalize and quantify earlier works.Comment: revised versio
Adiabatic Rotation, Quantum Search and Preparation of Superposition States
We introduce the idea of using adiabatic rotation to generate superpositions
of a large class of quantum states. For quantum computing this is an
interesting alternative to the well-studied "straight line" adiabatic
evolution. In ways that complement recent results, we show how to efficiently
prepare three types of states: Kitaev's toric code state, the cluster state of
the measurement-based computation model and the history state used in the
adiabatic simulation of quantum circuit. We also show that the method, when
adapted for quantum search, provides quadratic speedup as other optimal methods
do with the advantages that the problem Hamiltonian is time-independent and
that the energy gap above the ground state is strictly nondecreasing with time.
Likewise the method can be used for optimization as an alternative to the usual
adiabatic algorithm.Comment: Expanded to 6 pages; Accepted by Phys. Rev.
The Adiabatic Theorem for Quantum Systems with Spectral Degeneracy
By stating the adiabatic theorem of quantum mechanics in a clear and rigorous
way, we establish a necessary condition and a sufficient condition for its
validity, where the latter is obtained employing our recently developed
adiabatic perturbation theory. Also, we simplify further the sufficient
condition into a useful and simple practical test at the expenses of its
mathematical rigor. We present results for the most general case of quantum
systems, i.e., those with degenerate energy spectra. These conditions are of
upmost importance to assess the validity of practical implementations of
non-Abelian braiding and adiabatic quantum computation. To illustrate the
degenerate adiabatic approximation, and the necessary and sufficient conditions
for its validity, we analyze in depth an exactly solvable time-dependent
degenerate problem.Comment: 4 pages, no figures, RevTex4-1; v2: published versio
Adiabatic Product Expansion
The time-evolution operator for an explicitly time-dependent Hamiltonian is
expressed as the product of a sequence of unitary operators. These are obtained
by successive time-dependent unitary transformations of the Hilbert space
followed by the adiabatic approximation at each step. The resulting adiabatic
product expansion yields a generalization of the quantum adiabatic
approximation. Furthermore, it leads to an infinite class of exactly solvable
models.Comment: uuencoded LaTeX file, 8 page
Relativistic Comparison Theorems
Comparison theorems are established for the Dirac and Klein--Gordon
equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive
central potentials in d dimensions that support discrete Dirac eigenvalues
E^{(1)}_{k_d\nu} and E^{(2)}_{k_d\nu}. We prove that if V^{(1)}(r) \leq
V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered
E^{(1)}_{k_d\nu} \leq E^{(2)}_{k_d\nu}. This result generalizes an earlier more
restrictive theorem that required the wave functions to be node free. For the
the Klein--Gordon equation, similar reasoning also leads to a comparison
theorem provided in this case that the potentials are negative and the
eigenvalues are positive.Comment: 6 page
Quantum adiabatic optimization and combinatorial landscapes
In this paper we analyze the performance of the Quantum Adiabatic Evolution
algorithm on a variant of Satisfiability problem for an ensemble of random
graphs parametrized by the ratio of clauses to variables, . We
introduce a set of macroscopic parameters (landscapes) and put forward an
ansatz of universality for random bit flips. We then formulate the problem of
finding the smallest eigenvalue and the excitation gap as a statistical
mechanics problem. We use the so-called annealing approximation with a
refinement that a finite set of macroscopic variables (versus only energy) is
used, and are able to show the existence of a dynamic threshold
starting with some value of K -- the number of variables in
each clause. Beyond dynamic threshold, the algorithm should take exponentially
long time to find a solution. We compare the results for extended and
simplified sets of landscapes and provide numerical evidence in support of our
universality ansatz. We have been able to map the ensemble of random graphs
onto another ensemble with fluctuations significantly reduced. This enabled us
to obtain tight upper bounds on satisfiability transition and to recompute the
dynamical transition using the extended set of landscapes.Comment: 41 pages, 10 figures; added a paragraph on paper's organization to
the introduction, fixed reference
Adiabatic Perturbation Theory and Geometric Phases for Degenerate Systems
We introduce an adiabatic perturbation theory for quantum systems with
degenerate energy spectra. This perturbative series enables one to rigorously
establish conditions for the validity of the adiabatic theorem of quantum
mechanics for degenerate systems. The same formalism can be used to find
non-adiabatic corrections to the non-Abelian Wilczek-Zee geometric phase. These
corrections are relevant to assess the validity of the practical implementation
of the concept of fractional exchange statistics. We illustrate the formalism
by exactly solving a time-dependent problem and comparing its solution to the
perturbative one.Comment: 5 pages, no figures, RevTex4; v2: published versio
Adiabatic quantum state transfer in non-uniform triple-quantum-dot system
We introduce an adiabatic quantum state transfer scheme in a non-uniform
coupled triple-quantum-dot system. By adiabatically varying the external gate
voltage applied on the sender and receiver, the electron can be transferred
between them with high fidelity. By numerically solving the master equation for
a system with always-on interaction, it is indicated that the transfer fidelity
depends on the ration between the peak voltage and the maximum coupling
constants. The effect of coupling mismatch on the transfer fidelity is also
investigated and it is shown that there is a relatively large tolerance range
to permit high fidelity quantum state transfer.Comment: 6 pages, 5 figure
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