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Research Articles:*

**On the transport of
atomic ions in linear and multidimensional ion trap arrays**
(pp0501-0578) PDF

D.
Hucul, M. Yeo, W.K. Hensinger, J. Rabchuk, S. Olmschenk, and C. Monroe

Trapped atomic ions have become one of the most promising architectures
for a quantum computer, and current effort is now devoted to the
transport of trapped ions through complex segmented ion trap structures
in order to scale up to much larger numbers of trapped ion qubits. This
paper covers several important issues relevant to ion transport in any
type of complex multidimensional rf (Paul) ion trap array. We develop a
general theoretical framework for the application of time-dependent
electric fields to shuttle laser-cooled ions along any desired
trajectory, and describe a method for determining the effect of
arbitrary shuttling schedules on the quantum state of trapped ion
motion. In addition to the general case of linear shuttling over short
distances, we introduce issues particular to the shuttling through
multidimensional junctions, which are required for the arbitrary control
of the positions of large arrays of trapped ions. This includes the
transport of ions around a corner, through a cross or T junction, and
the swapping of positions of multiple ions in a laser-cooled crystal.
Where possible, we make connections to recent experimental results in a
multidimensional T junction trap, where arbitrary 2-dimensional
transport was realized.

**On solving systems of
random linear disequations**
(pp0579-0594) PDF

G.
Ivanyos

An important special case of the hidden subgroup problem is equivalent
to the hidden shift problem over abelian groups. An efficient solution
to the latter problem could serve as a building block of quantum hidden
subgroup algorithms over solvable groups. The main idea of a promising
approach to the hidden shift problem is a reduction to solving systems
of certain random disequations in finite abelian groups. By a
disequation we mean a constraint of the form $f(x)\neq 0$. In our case,
the functions on the left hand side are generalizations of linear
functions. The input is a random sample of functions according to a
distribution which is up to a constant factor uniform over the "linear"
functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown
element $u\in A$. The goal is to find $u$, or, more precisely, all the
elements $u'\in A$ satisfying the same disequations as $u$. In this
paper we give a classical probabilistic algorithm which solves the
problem in an abelian $p$-group $A$ in time polynomial in the sample
size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of
$A$.

**Homological invariants
of stabilizer states**
(pp0585-0621) PDF

K.
Wirthmuller

We propose a new kind of invariant of multi-party stabilizer states with
respect to local Clifford equivalence. These homological invariants are
discrete entities defined in terms of the entanglement a state enjoys
with respect to arbitrary groupings of the parties, and they may be
thought of as reflecting entanglement in a qualitative way. We
investigate basic properties of the invariants and link them with known
results on the extraction of GHZ states.

**Faked states attack
using detector efficiency mismatch on SARG04, phase-time, DPSK, and
Ekert protocols**
(pp0622-0635) PDF

V.
Makarov and J. Skaar

In quantum cryptosystems, variations in detector efficiency can be
exploited to stage a successful attack. This happens when the
efficiencies of Bob's two detectors are different functions of a control
parameter accessible to Eve (e.g., timing of the incoming pulses). It
has previously been shown that the Bennett-Brassard 1984 (BB84) protocol
is vulnerable to this attack. In this paper, we show that several other
protocols and encodings may also be vulnerable. We consider a faked
states attack in the case of a partial efficiency mismatch on the
Scarani-Acin-Ribordy-Gisin 2004 (SARG04) protocol, and derive the
quantum bit error rate as a function of detector efficiencies.
Additionally, it is shown how faked states can in principle be
constructed for quantum cryptosystems that use a phase-time encoding,
the differential phase shift keying (DPSK) and the Ekert protocols.

**A fast quantum circuit
for addition with few qubits**
(pp0636-0649) PDF

Y.
Takahashi and N. Kunihiro

We show how to construct a fast quantum circuit for computing the sum of
two $n$-bit binary numbers with few qubits. The constructed circuit uses
$O(n/\log n)$ ancillary qubits and its depth and size are $O(\log n)$
and $O(n)$, respectively. The number of ancillary qubits is
asymptotically less than that in Draper et al.'s quantum carry-lookahead
adder, and the depth and size are asymptotically the same as those of
Draper et al.'s. Moreover, we show that the circuit is useful for
constructing an efficient quantum circuit for Shor's factoring
algorithm.

**PEPS as unique ground
states of local Hamiltonians**
(pp0650-0663) PDF

D.
Perez-Garcia, F. Verstraete, J.I. Cirac, and M.M. Wolf

In this paper we consider projected entangled pair states (PEPS) on
arbitrary lattices. We construct local parent Hamiltonians for each PEPS
and isolate a condition under which the state is the unique ground state
of the Hamiltonian. This condition, verified by generic PEPS and
examples like the AKLT model, is an injective relation between the
boundary and the bulk of any local region. While it implies the
existence of an energy gap in the 1D case we will show that in certain
cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be
gapless with a critical ground state. To show this we invoke a mapping
between classical and quantum models and prove that in these cases the
injectivity relation between boundary and bulk solely depends on the
lattice geometry.

**On separability of
graphs with some entangled edges**
(pp0664-0670) PDF

H.
Rahiminia and M. Amini

In this paper we give a combinatorial proof of the entanglement of a
graph whose number of vertices is a product of two primes, and has
exactly one entangled edge, or all of whose entangled edges are
emanating from one vertex. Thereby, we provide easier and more direct
proofs of the conjecture by Braunstein, Ghosh, and Severini in {\it
quant-ph}/0406165 v2 (2006) [2].

**Matrix rearrangement
approach for the entangling power with hybrid qudit systems**
(pp0671-0680) PDF

X.-M.
Lu, X. Wang, Y. Yang, and J. Chen

We extend the former matrix rearrangement approach of the entangling
power to the general cases, without requirement of same dimensions of
the subsystems. The entangling power of a unitary operator is completely
determined by its realignment and partial transposition. As
applications, we calculate the entangling power for the Ising
interaction and the isotropic Heisenberg interaction in the hybrid qudit
$d_1 \times d_2$ systems.

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