2,615 research outputs found
Carmichael Numbers on a Quantum Computer
We present a quantum probabilistic algorithm which tests with a polynomial
computational complexity whether a given composite number is of the Carmichael
type. We also suggest a quantum algorithm which could verify a conjecture by
Pomerance, Selfridge and Wagstaff concerning the asymptotic distribution of
Carmichael numbers smaller than a given integer.Comment: 7 pages, Latex/REVTEX fil
Quantum Computers and Unstructured Search: Finding and Counting Items with an Arbitrarily Entangled Initial State
Grover's quantum algorithm for an unstructured search problem and the Count
algorithm by Brassard et al. are generalized to the case when the initial state
is arbitrarily and maximally entangled. This ansatz might be relevant with
quantum subroutines, when the computational qubits and the environment are
coupled, and in general when the control over the quantum system is partial.Comment: Completely revised version, accepted for publ. on PLA.; 11 page
Optimal phase estimation and square root measurement
We present an optimal strategy having finite outcomes for estimating a single
parameter of the displacement operator on an arbitrary finite dimensional
system using a finite number of identical samples. Assuming the uniform {\it a
priori} distribution for the displacement parameter, an optimal strategy can be
constructed by making the {\it square root measurement} based on uniformly
distributed sample points. This type of measurement automatically ensures the
global maximality of the figure of merit, that is, the so called average score
or fidelity. Quantum circuit implementations for the optimal strategies are
provided in the case of a two dimensional system.Comment: Latex, 5 figure
Time-optimal Unitary Operations in Ising Chains II: Unequal Couplings and Fixed Fidelity
We analytically determine the minimal time and the optimal control laws
required for the realization, up to an assigned fidelity and with a fixed
energy available, of entangling quantum gates () between
indirectly coupled qubits of a trilinear Ising chain. The control is coherent
and open loop, and it is represented by a local and continuous magnetic field
acting on the intermediate qubit. The time cost of this local quantum operation
is not restricted to be zero. When the matching with the target gate is perfect
(fidelity equal to one) we provide exact solutions for the case of equal Ising
coupling. For the more general case when some error is tolerated (fidelity
smaller than one) we give perturbative solutions for unequal couplings.
Comparison with previous numerical solutions for the minimal time to generate
the same gates with the same Ising Hamiltonian but with instantaneous local
controls shows that the latter are not time-optimal.Comment: 11 pages, no figure
Time machines and the Principle of Self-Consistency as a consequence of the Principle of Stationary Action (II): the Cauchy problem for a self-interacting relativistic particle
We consider the action principle to derive the classical, relativistic motion
of a self-interacting particle in a 4-D Lorentzian spacetime containing a
wormhole and which allows the existence of closed time-like curves. In
particular, we study the case of a pointlike particle subject to a
`hard-sphere' self-interaction potential and which can traverse the wormhole an
arbitrary number of times, and show that the only possible trajectories for
which the classical action is stationary are those which are globally
self-consistent. Generically, the multiplicity of these trajectories (defined
as the number of self-consistent solutions to the equations of motion beginning
with given Cauchy data) is finite, and it becomes infinite if certain
constraints on the same initial data are satisfied. This confirms the previous
conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and
Thorne that the Cauchy initial value problem in the presence of a wormhole
`time machine' is classically `ill-posed' (far too many solutions). Our results
further extend the recent claim by Novikov et al. that the `Principle of
self-consistency' is a natural consequence of the `Principle of minimal
action.'Comment: 39 pages, latex fil
Star configuration points and plane curves
2siLet â1,...,â1 be l lines in â2 such that no three lines meet in a point. Let X(l) be the set of points {âi ⊠âj {divides} 1 ⤠i < j ⤠l} â â2. We call X(l) a star configuration. We describe all pairs (d, l) such that the generic degree d curve in â2 contains an X(l). Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems. Š 2011 American Mathematical Society.openopenCarlini E.; van Tuyl A.Carlini, E.; van Tuyl, A
Brachistochrone of Entanglement for Spin Chains
We analytically investigate the role of entanglement in time-optimal state
evolution as an appli- cation of the quantum brachistochrone, a general method
for obtaining the optimal time-dependent Hamiltonian for reaching a target
quantum state. As a model, we treat two qubits indirectly cou- pled through an
intermediate qubit that is directly controllable, which represents a typical
situation in quantum information processing. We find the time-optimal unitary
evolution law and quantify residual entanglement by the two-tangle between the
indirectly coupled qubits, for all possible sets of initial pure quantum states
of a tripartite system. The integrals of the motion of the brachistochrone are
determined by fixing the minimal time at which the residual entanglement is
maximized. Entan- glement plays a role for W and GHZ initial quantum states,
and for the bi-separable initial state in which the indirectly coupled qubits
have a nonzero value of the 2-tangle.Comment: 9 pages, 4 figure
Quantum Brachistochrone for Mixed States
We present a general formalism based on the variational principle for finding
the time-optimal quantum evolution of mixed states governed by a master
equation, when the Hamiltonian and the Lindblad operators are subject to
certain constraints. The problem reduces to solving first a fundamental
equation (the {\it quantum brachistochrone}) for the Hamiltonian, which can be
written down once the constraints are specified, and then solving the
constraints and the master equation for the Lindblad and the density operators.
As an application of our formalism, we study a simple one-qubit model where the
optimal Lindblad operators control decoherence and can be simulated by a
tunable coupling with an ancillary qubit. It is found that the evolution
through mixed states can be more efficient than the unitary evolution between
given pure states. We also discuss the mixed state evolution as a finite time
unitary evolution of the system plus an environment followed by a single
measurement. For the simplest choice of the constraints, the optimal duration
time for the evolution is an exponentially decreasing function of the
environment's degrees of freedom.Comment: 8 pages, 3 figure
The vacuum polarization around an axionic stringy black hole
We consider the effect of vacuum polarization around the horizon of a 4
dimensional axionic stringy black hole. In the extreme degenerate limit
(), the lower limit on the black hole mass for avoiding the polarization
of the surrounding medium is ( is the
proton mass), according to the assumed value of the axion mass (). In this case, there are no upper bounds on the mass
due to the absence of the thermal radiation by the black hole. In the
nondegenerate (classically unstable) limit (), the black hole always
polarizes the surrounding vacuum, unless the effective cosmological constant of
the effective stringy action diverges.Comment: 7 pages, phyzzx.tex, ROM2F-92-3
On the Stability of a Stringy Black Hole
We study the stability under perturbations of a charged four dimensional
stringy black hole arising from gauging a previously studied WZW model. We find
that the black hole is stable only in the extremal case .Comment: 14 pages + 1 figure (not included but available on request
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