19 research outputs found
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
A quantum system will stay near its instantaneous ground state if the
Hamiltonian that governs its evolution varies slowly enough. This quantum
adiabatic behavior is the basis of a new class of algorithms for quantum
computing. We test one such algorithm by applying it to randomly generated,
hard, instances of an NP-complete problem. For the small examples that we can
simulate, the quantum adiabatic algorithm works well, and provides evidence
that quantum computers (if large ones can be built) may be able to outperform
ordinary computers on hard sets of instances of NP-complete problems.Comment: 15 pages, 6 figures, email correspondence to [email protected] ; a
shorter version of this article appeared in the April 20, 2001 issue of
Science; see http://www.sciencemag.org/cgi/content/full/292/5516/47
Improved Error-Scaling for Adiabatic Quantum State Transfer
We present a technique that dramatically improves the accuracy of adiabatic
state transfer for a broad class of realistic Hamiltonians. For some systems,
the total error scaling can be quadratically reduced at a fixed maximum
transfer rate. These improvements rely only on the judicious choice of the
total evolution time. Our technique is error-robust, and hence applicable to
existing experiments utilizing adiabatic passage. We give two examples as
proofs-of-principle, showing quadratic error reductions for an adiabatic search
algorithm and a tunable two-qubit quantum logic gate.Comment: 10 Pages, 4 figures. Comments are welcome. Version substantially
revised to generalize results to cases where several derivatives of the
Hamiltonian are zero on the boundar
Quantum search by measurement
We propose a quantum algorithm for solving combinatorial search problems that
uses only a sequence of measurements. The algorithm is similar in spirit to
quantum computation by adiabatic evolution, in that the goal is to remain in
the ground state of a time-varying Hamiltonian. Indeed, we show that the
running times of the two algorithms are closely related. We also show how to
achieve the quadratic speedup for Grover's unstructured search problem with
only two measurements. Finally, we discuss some similarities and differences
between the adiabatic and measurement algorithms.Comment: 8 pages, 2 figure
Cooling atoms into entangled states
We discuss the possibility of preparing highly entangled states by simply
cooling atoms into the ground state of an applied interaction Hamiltonian. As
in laser sideband cooling, we take advantage of a relatively large detuning of
the desired state, while all other qubit states experience resonant laser
driving. Once spontaneous emission from excited atomic states prepares the
system in its ground state, it remains there with a very high fidelity for a
wide range of experimental parameters and all possible initial states. After
presenting the general theory, we discuss concrete applications with one and
two qubits.Comment: 16 pages, 6 figures, typos correcte
Effective Hamiltonian approach to adiabatic approximation in open systems
The adiabatic approximation in open systems is formulated through the
effective Hamiltonian approach. By introducing an ancilla, we embed the open
system dynamics into a non-Hermitian quantum dynamics of a composite system,
the adiabatic evolution of the open system is then defined as the adiabatic
dynamics of the composite system. Validity and invalidity conditions for this
approximation are established and discussed. A High-order adiabatic
approximation for open systems is introduced. As an example, the adiabatic
condition for an open spin- particle in time-dependent magnetic
fields is analyzed.Comment: 6 pages, 2 figure
Robustness of adiabatic passage trough a quantum phase transition
We analyze the crossing of a quantum critical point based on exact results
for the transverse XY model. In dependence of the change rate of the driving
field, the evolution of the ground state is studied while the transverse
magnetic field is tuned through the critical point with a linear ramping. The
excitation probability is obtained exactly and is compared to previous studies
and to the Landau-Zener formula, a long time solution for non-adiabatic
transitions in two-level systems. The exact time dependence of the excitations
density in the system allows to identify the adiabatic and diabatic regions
during the sweep and to study the mesoscopic fluctuations of the excitations.
The effect of white noise is investigated, where the critical point transmutes
into a non-hermitian ``degenerate region''. Besides an overall increase of the
excitations during and at the end of the sweep, the most destructive effect of
the noise is the decay of the state purity that is enhanced by the passage
through the degenerate region.Comment: 16 pages, 15 figure
Convergence theorems for quantum annealing
We prove several theorems to give sufficient conditions for convergence of
quantum annealing, which is a protocol to solve generic optimization problems
by quantum dynamics. In particular the property of strong ergodicity is proved
for the path-integral Monte Carlo implementation of quantum annealing for the
transverse Ising model under a power decay of the transverse field. This result
is to be compared with the much slower inverse-log decay of temperature in the
conventional simulated annealing. Similar results are proved for the Green's
function Monte Carlo approach. Optimization problems in continuous space of
particle configurations are also discussed.Comment: 19 page