Vol.7
No.4
May 1, 2007
Review:
Quantum computaton from a
quantum logical perspective (pp281296) PDF
J.
Bub
It is wellknown that Shor's factorization algorithm,
Simon's periodfinding algorithm, and Deutsch's original XOR algorithm
can all be formulated as solutions to a hidden subgroup problem. Here
the salient features of the informationprocessing in the three
algorithms are presented from a different perspective, in terms of the
way in which the algorithms exploit the nonBoolean quantum logic
represented by the projective geometry of Hilbert space. From this
quantum logical perspective, the XOR algorithm appears directly as a
special case of Simon's algorithm, and all three algorithms can be seen
as exploiting the nonBoolean logic represented by the subspace
structure of Hilbert space in a similar way. Essentially, a global
property of a function (such as a period, or a disjunctive property) is
encoded as a subspace in Hilbert space representing a quantum
proposition, which can then be efficiently distinguished from
alternative propositions, corresponding to alternative global
properties, by a measurement (or sequence of measurements) that
identifies the target proposition as the proposition represented by the
subspace containing the final state produced by the algorithm.
Research Articles:
Noise threshold for a faulttolerant
twodimensional lattice architecture (pp297318) PDF
K.M.
Svore, D.P. DiVincenzo, and B.M. Terhal
We consider a model of quantum computation in which the
set of operations is limited to nearestneighbor interactions on a 2D
lattice. We model movement of qubits with noisy \SWAP\ operations. For
this architecture we design a faulttolerant coding scheme using the
concatenated $[[7,1,3]]$ Steane code. Our scheme is potentially
applicable to iontrap and solidstate quantum technologies. We
calculate a lower bound on the noise threshold for our local model using
a detailed failure probability analysis. We obtain a threshold of $1.85
\times 10^{5}$ for the local setting, where memory error rates are
onetenth of the failure rates of gates, measurement, and preparation
steps. For the analogous nonlocal setting, we obtain a noise threshold
of $3.61 \times 10^{5}$. Our results thus show that the additional
\SWAP\ operations required to move qubits in the local model affect the
noise threshold only moderately.
Winning
strategies for pseudotelepathy games using single nonlocal box (pp319328) PDF
S.
Kunkri, G. Kar, S. Ghosh, and A. Roy
Using a single NLbox, a winning strategy is given for
the impossible colouring pseudotelepathy game for the set of vectors
having KochenSpecker property in four dimension. A sufficient condition
given regarding the structure of the impossible colouring
pseudotelepathy game for general $d$dimension. A winning strategy for
this game is then described with single use of NLbox.
A simple participant attack on the BradlerDusek protocol (pp329334) PDF
F.
Gao, S. Qin, Q. Wen, and F. Zhu
The ringarrangement quantum secret sharing protocol in
the paper [K. Br\'{a}dler and M. Du\v{s}ek (2004) {\textit{J. Opt. B:
Quantum Semiclass. Opt.}} {\textbf{6}} 63] is analyzed and it is shown
that this protocol is secure for any eavesdropper except for a dishonest
participant. For example, by a special strategy, Bob can steal Charlie's
portion of information without being detected and then recover Alice's
secret by himself. We give a description of this strategy and point out
a possible way to improve the protocol to stand against this attack.
Computational complexity of the quantum separability problem (pp335370) PDF
L.M.
Ioannou
Ever since entanglement was identified as a computational
and cryptographic resource, researchers have sought efficient ways to
tell whether a given density matrix represents an unentangled, or
\emph{separable}, state. This paper gives the first systematic and
comprehensive treatment of this (bipartite) quantum separability
problem, focusing on its deterministic (as opposed to randomized)
computational complexity. First, I review the onesided tests for
separability, paying particular attention to the semidefinite
programming methods. Then, I discuss various ways of formulating the
quantum separability problem, from exact to approximate formulations,
the latter of which are the paper's main focus. I then give a thorough
treatment of the problem's relationship with NP, NPcompleteness, and
coNP. I also discuss extensions of Gurvits' NPhardness result to
strong NPhardness of certain related problems. A major open question is
whether the NPcontained formulation (QSEP) of the quantum separability
problem is KarpNPcomplete; QSEP may be the first natural example of a
problem that is TuringNPcomplete but not KarpNPcomplete. Finally, I
survey all the proposed (deterministic) algorithms for the quantum
separability problem, including the bounded search for symmetric
extensions (via semidefinite programming), based on the recent quantum
de Finetti theorem \cite{DPS02,DPS04,qphCKMR06}; and the
entanglementwitness search (via interiorpoint algorithms and global
optimization) \cite{ITCE04,IT06}. These two algorithms have the lowest
complexity, with the latter being the best under advice of
asymptotically optimal pointcoverings of the sphere.
Mutually unbiased bases and orthogonal decompositions of Lie algebras (pp371382) PDF
P.O.
Boykin, M. Sitharam, P.H. Tiep, and P. Wocjan
We establish a connection between the problem of
constructing maximal collections of mutually unbiased bases (MUBs) and
an open problem in the theory of Lie algebras. More precisely, we show
that a collection of $\mu$ MUBs in $\K^n$ gives rise to a collection of
$\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$
that are pairwise orthogonal with respect to the Killing form, where
$\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in
$\C^n$ gives rise to a socalled orthogonal decomposition (OD) of $sl_n(\C)$.
The converse holds if the Cartan subalgebras in the OD are also $\dag$closed,
i.e., closed under the adjoint operation. In this case, the Cartan
subalgebras have unitary bases, and the above correspondence becomes
equivalent to a result of \cite{bbrv02} relating collections of MUBs to
collections of maximal commuting classes of unitary error bases, i.e.,
orthogonal unitary matrices. This connection implies that for $n\le 5$
an essentially unique complete collection of MUBs exists. We define \emph{monomial
MUBs}, a class of which all known MUB constructions are members, and use
the above connection to show that for $n=6$ there are at most three
monomial MUBs.
The quantum Fourier transform on a linear nearest neighbor architecture (pp383391) PDF
Y.
Takahashi, N. Kunihiro, and K. Ohta
We show how to construct an efficient quantum circuit for
computing a good approximation of the quantum Fourier transform on a
linear nearest neighbor architecture. The constructed circuit uses no
ancillary qubits and its depth and size are $O(n)$ and $O(n\log n)$,
respectively, where $n$ is the length of the input. The circuit is
useful for decreasing the size of Fowler et al.'s quantum circuit for
Shor's factoring algorithm on a linear nearest neighbor architecture.
General classes of impossible operations through the existence of
incomparable states (pp392400) PDF
I.
Chattopadhyay and D. Sarkar
In this work we show that the most general class of
antiunitary operators are nonphysical in nature through the existence
of incomparable pure bipartite entangled states. It is also shown that a
large class of innerproductpreserving operations defined only on the
three qubits having spindirections along x,y and z are impossible. If
we perform such an operation locally on a particular pure bipartite
state then it will exactly transform to another pure bipartite state
that is incomparable with the original one. As subcases of the above
results we find the nonphysical nature of universal exact flipping
operation and existence of universal Hadamard gate. Beyond the
information conservation in terms of entanglement, this work shows how
an impossible local operation evolve with the joint system in a
nonphysical way.
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