We examine the validity of the harmonic approximation, where the
radio-frequency ion trap is treated as a harmonic trap, in the problem
regarding the controlled collision of a trapped atom and a single trapped ion.
This is equivalent to studying the effect of the micromotion since this motion
must be neglected for the trapped ion to be considered as a harmonic
oscillator. By applying the transformation of Cook and Shankland we find that
the micromotion can be represented by two periodically oscillating operators.
In order to investigate the effect of the micromotion on the dynamics of a
trapped atom-ion system, we calculate (i) the coupling strengths of the
micromotion operators by numerical integration and (ii) the quasienergies of
the system by applying the Floquet formalism, a useful framework for studying
periodic systems. It turns out that the micromotion is not negligible when the
distance between the atom and the ion traps is shorter than a characteristic
distance. Within this range the energy diagram of the system changes remarkably
when the micromotion is taken into account, which leads to undesirable
consequences for applications that are based on an adiabatic process of the
trapped atom-ion system. We suggest a simple scheme for bypassing the
micromotion effect in order to successfully implement a quantum controlled
phase gate proposed previously and create an atom-ion macromolecule. The
methods presented here are not restricted to trapped atom-ion systems and can
be readily applied to studying the micromotion effect in any system involving a
single trapped ion.Comment: 25 pages, 8 figure

The search for "a quantum needle in a quantum haystack" is a metaphor for the problem of finding out which one of a permissible set of unitary mappings-the oracles-is implemented by a given black box. Grover's algorithm solves this problem with quadratic speed-up as compared with the analogous search for "a classical needle in a classical haystack." Since the outcome of Grover's algorithm is probabilistic-it gives the correct answer with high probability, not with certainty-the answer requires verification. For this purpose we introduce specific test states, one for each oracle. These test states can also be used to realize "a classical search for the quantum needle" which is deterministic-it always gives a definite answer after a finite number of steps-and faster by a factor of 3.41 than the purely classical search. Since the test-state search and Grover's algorithm look for the same quantum needle, the average number of oracle queries of the test-state search is the classical benchmark for Grover's algorithm.

Complexity is often invoked alongside size and mass as a characteristic of
macroscopic quantum objects. In 2004, Aaronson introduced the \textit{tree
size} (TS) as a computable measure of complexity and studied its basic
properties. In this paper, we improve and expand on those initial results. In
particular, we give explicit characterizations of a family of states with
superpolynomial complexity $n^{\Omega(\log n)}= \mathrm{TS} =O(\sqrt{n}!)$ in
the number of qubits $n$; and we show that any matrix-product state whose
tensors are of dimension $D\times D$ has polynomial complexity
$\mathrm{TS}=O(n^{\log_2 2D})$.Comment: 7 pages, 2 figure

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