Vol.4 No.3 May 30,
2004
Research and
Review Articles: Scalable
trapped ion quantum computation with a probabilistic ionphoton mapping
(pp165173)
L.M. Duan, B.B. Blinov, D.L.
Moehring and C. Monroe
We propose a method for scaling trapped ions for largescale quantum
computation and communication based on a probabilistic ionphoton
mapping. Deterministic quantum gates between remotely located trapped
ions can be achieved through detection of spontaneouslyemitted photons,
accompanied by the local Coulomb interaction between neighboring ions.
We discuss gate speeds and tolerance to experimental noise for different
probabilistic entanglement schemes.
An exact effective
twoqubit gate in a chain of three spins (pp174185)
M.H. Yung, D.W. Leung and S. Bose
We show that an effective twoqubit gate can be obtained from the free
evolution of three spins in a chain with nearest neighbor XY coupling,
without local manipulations. This gate acts on the two remote spins and
leaves the mediating spin unchanged. It can be used to perfectly
transfer an arbitrary quantum state from the first spin to the last spin
or to simultaneously communicate one classical bit in each direction.
One ebit can be generated in half of the time for state transfer. For
longer spin chains, we present methods to create or transfer
entanglement between the two end spins in half of the time required for
quantum state transfer, given tunable coupling strength and local
magnetic field. We also examine imperfect state transfer through a
homogeneous XY chain.
Quantum logical
networks for probabilistic teleportation of many particle state of
general form (pp186195)
T. Gao, F.L. Yan and Z.X. Wang
The scheme for probabilistic teleportation of an $N$particle state of
general form is proposed. As the special cases we construct efficient
quantum logic networks for implementing probabilistic teleportation of a
twoparticle state, a threeparticle state and a fourparticle state of
general form, built from single qubit gates, twoqubit controllednot
gates, Von Neumann measurement and classically controlled operations.
Entanglement
concentration of individual photon pairs via linear optical logic
(pp196200)
C.W. Zhang
We propose a scheme for concentrating nonmaximally pure and mixed
polarizationentangled state of individual photon pairs. The scheme uses
only simple linear optical elements and may be feasible within current
optical technology.
How significant are
the known collision and element distinctness quantum algorithms?
(pp201206)
L. Grover and T. Rudolph
Quantum search is a technique for searching $N$ possibilities for a
desired target in $O(\sqrt{N})$ steps. It has been applied in the design
of quantum algorithms for several structured problems. Many of these
algorithms require significant amount of quantum hardware. In this paper
we propose the criterion that an algorithm which requires $O(S)$
hardware should be considered significant if it produces a speedup of
better than $O\left(\sqrt{S}\right)$ over a simple quantum search
algorithm. This is because a speedup of $O\left(\sqrt{S}\right)$ can be
trivially obtained by dividing the search space into $S$ separate parts
and handing the problem to $S$ independent processors that do a quantum
search (in this paper we drop all logarithmic factors when discussing
time/space complexity). Known algorithms for collision and element
distinctness exactly saturate the criterion.
Simplifying Schmidt
number witnesses via higherdimensional embeddings (pp207221)
F. Hulpke, D. Bruss, M. Levenstein
and A. Sanpera
We apply the generalised concept of witness operators to arbitrary
convex sets, and review the criteria for the optimisation of these
general witnesses. We then define an embedding of state vectors and
operators into a higherdimensional Hilbert space. This embedding leads
to a connection between any Schmidt number witness in the original
Hilbert space and a witness for Schmidt number two (i.e. the most
general entanglement witness) in the appropriate enlarged Hilbert space.
Using this relation we arrive at a conceptually simple method for the
construction of Schmidt number witnesses in bipartite systems.
An upper bound on
the threshold quantum decoherence rate (pp222228)
A.A. Razborov
Let $\eta_0$ be the supremum of those $\eta$ for which every polysize
quantum circuit can be simulated by another polysize quantum circuit
with gates of fanin $\leq 2$ that tolerates random noise independently
occurring on all wires at the constant rate $\eta$. Recent fundamental
results showing the principal fact $\eta_0>0$ give estimates like
$\eta_0\geq 10^{6}\mbox{}10^{4}$, whereas the only upper bound known
before is $\eta_0\leq 0.74$.}{In this note we improve the latter bound
to $\eta_0\leq 1/2$, under the assumption ${\bf QP}\not\subseteq {\bf
QNC^1}$. More generally, we show that if the decoherence rate $\eta$ is
greater than 1/2, then we can not even store a single qubit for more
than logarithmic time. Our bound also generalizes to the simulating
circuits allowing gates of any (constant) fanin $k$, in which case we
have $\eta_0\leq 1\frac 1k$.
Quantum solution to
the hidden subgroup problem for PolyNearHamiltonian groups
(pp229235)
D. Gavinsky
The Hidden Subgroup Problem (HSP) has been widely studied in the
context of quantum computing and is known to be efficiently solvable for
Abelian groups, yet appears to be difficult for many nonAbelian ones.
An efficient algorithm for the HSP over a group \f G\ runs in
time polynomial in \f{n\deq\logG.} For any subgroup \f H\ of \f G, let
\f{N(H)} denote the normalizer of \f H. Let \MG\ denote the intersection
of all normalizers in \f G (i.e., \f{\MG=\cap_{H\leq G}N(H)}). \MG\ is
always a subgroup of \f G and the index \f{[G:\MG]} can be taken as a
measure of ``how nonAbelian'' \f G is (\f{[G:\MG] = 1} for Abelian
groups). This measure was considered by Grigni, Schulman, Vazirani and
Vazirani, who showed that whenever \f{[G:\MG]\in\exp(O(\log^{1/2}n))}
the corresponding HSP can be solved efficiently (under certain
assumptions). We show that whenever \f{[G:\MG]\in\poly(n)} the
corresponding HSP can be solved efficiently, under the same assumptions
(actually, we solve a slightly more general case of the HSP and also
show that some assumptions may be relaxed).
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