Vol.4 No.4 July 05,
2004
Research and
Review Articles:
Implementation of Shor's
algorithm
on a linear nearest neighbour qubit array
(pp237251)
A.G.
Fowler, S.J.
Devitt and L.C.L.
Hollenberg
Shor's algorithm, which given appropriate hardware can factorise an
integer N in a time polynomial in its binary length L, has
arguably spurred the race to build a practical quantum computer. Several
different quantum circuits implementing Shor's algorithm have been
designed, but each tacitly assumes that arbitrary pairs of qubits within
the computer can be interacted. While some quantum computer
architectures possess this property, many promising proposals are best
suited to realising a single line of qubits with nearest neighbour
interactions only. In light of this, we present a circuit implementing
Shor's factorisation algorithm designed for such a linear nearest
neighbour architecture. Despite the interaction restrictions, the
circuit requires just 2L+4 qubits and to leading order requires
8L^4 2qubit gates arranged in a circuit of depth 32L^3
 identical to leading order to that possible using an architecture
that can interact arbitrary pairs of qubits.
Connections between
relative entropy of entanglement and geometric measure of entanglement
(pp252272)
T.C.
Wei, M. Ericsson, P.M.
Goldbart and W.J.
Munro
As two of the most important entanglement measuresthe entanglement of
formation and the entanglement of distillationhave so far been
limited to bipartite settings, the study of other entanglement measures
for multipartite systems appears necessary. Here, connections between
two other entanglement measuresthe relative entropy of entanglement
and the geometric measure of entanglementare investigated. It is
found that for arbitrary pure states the latter gives rise to a lower
bound on the former. For certain pure states, some bipartite and some
multipartite, this lower bound is saturated, and thus their relative
entropy of entanglement can be found analytically in terms of their
known geometric measure of entanglement. For certain mixed states, upper
bounds on the relative entropy of entanglement are also established.
Numerical evidence strongly suggests that these upper bounds are tight,
i.e., they are actually the relative entropy of entanglement.
Self testing
quantum apparatus
(pp273286)
D.
Mayers and A. Yao
We study, in the context of quantum information and quantum
communication, a configuration of devices that includes (1) a source of
some unknown bipartite quantum state that is claimed to be the Bell
state $\Phi^+$ and (2) two spatially separated but otherwise unknown
measurement apparatus, one on each side, that are each claimed to
execute an orthogonal measurement at an angle $\theta \in \{\pi/8, 0,
\pi/8\}$ that is chosen by the user. We show that, if the nine distinct
probability distributions that are generated by the self checking
configuration, one for each pair of angles, are consistent with the
specifications, the source and the two measurement apparatus are
guaranteed to be identical to the claimed specifications up to a local
change of basis on each side. We discuss the connection with quantum
cryptography.
Cluster states,
algorithms and graphs
(pp287324)
D. Schlingemann
The present paper is concerned with the concept of the oneway quantum
computer, beyond binarysystems, and its relation to the concept of
stabilizer quantum codes. This relation is exploited to analyze a
particular class of quantum algorithms, called graph algorithms, which
correspond in the binary case to the Clifford group part of a network
and which can efficiently be implemented on a oneway quantum computer.
These algorithms can ``completely be solved" in the sense that the
manipulation of quantum states in each step can be computed explicitly.
Graph algorithms are precisely those which implement encoding schemes
for graph codes. Starting from a given initial graph, which represents
the underlying resource of multipartite entanglement, each step of the
algorithm is related to a explicit transformation on the graph.
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