Vol.5 No.6
September 1,
2005
Research Articles:
Surfaceelectrode architecture for
iontrap quantum information processing (pp419439)
J. Chiaverini, R.B. Blakestad, J.
Britton, J.D. Jost, C. Langer, D. Leibfried, R. Ozeri and D.J. Wineland
We investigate a surfacemounted electrode geometry for
miniature linear radio frequency Paul ion
traps. The electrodes reside in a single plane
on a substrate, and the pseudopotential minimum of the
trap is located above the substrate at a distance on the order of
the electrodes' lateral extent or separation. This architecture
provides the possibility to apply standard microfabrication
principles to the construction of multiplexed ion traps, which
may be of particular importance in light of
recent proposals for largescale quantum
computation based on individual trapped ions.
A linearsize
quantum circuit for addition with no ancillary qubits
(pp440448)
Y. Takahashi and N. Kunihiro
We construct a quantum circuit for addition of two
$n$bit binary numbers that uses no ancillary
qubits. The circuit is based on the
ripplecarry approach. The depth and size of the circuit are
$O(n)$. This is an affirmative answer to the question of Kutin
\cite{Kutin} as to whether a lineardepth quantum circuit for
addition can be constructed without ancillary
qubits using the ripplecarry approach. We
also construct quantum circuits for addition modulo $2^n$,
subtraction, and comparison that use no ancillary qubits by
modifying the circuit for addition.
Two
slightlyentangled NPcomplete problems (pp449455)
R. Orus
We perform a mathematical analysis of the
classical computational complexity of two
genuine quantummechanical problems, which are
inspired in the calculation of the expected magnetizations and the
entanglement between subsystems for a quantum spin system. These
problems, which we respectively call SES and
SESSP, are specified in terms of pure slightlyentangled
quantum states of $n$ qubits, and rigorous mathematical
proofs that they belong to the NPComplete complexity class are
presented. Both SES and SESSP are, therefore, computationally equivalent
to the relevant $3$SAT problem, for which an efficient algorithm is yet
to be discovered.
Secure assisted
quantum computation (pp456466)
A.M. Childs
Suppose Alice wants to perform some computation
that could be done quickly on a quantum
computer, but she cannot do universal quantum
computation. Bob can do universal quantum computation and claims he
is willing to help, but Alice wants to be sure that Bob cannot
learn her input, the result of her
calculation, or perhaps even the function she
is trying to compute. We describe a simple, efficient protocol by
which Bob can help Alice perform the computation, but there is no
way for him to learn anything about it. We
also discuss techniques for Alice to detect
whether Bob is honestly helping her or if he is
introducing errors.
Transformation of
quantum states using uniformly controlled rotations (pp467473)
M. Mottonen, J. J. Vartiainen, V.
Bergholm and M. M. Salomaa
We consider a unitary transformation which maps any
given pure state of an $n$qubit quantum
register into another one. This transformation
has applications in the initialization of a quantum
computer, and also in some quantum algorithms.
Employing uniformly controlled rotations, we present a quantum
circuit of $2^{n+2}4n4$ CNOT gates and
$2^{n+2}5$ onequbit elementary rotations
that effects the state transformation. The
complexity of the circuit is noticeably lower than the previously
published results. Moreover, we present an analytic expression
for the rotation angles needed for the
transformation.
Optimized quantum
implementation of elliptic curve arithmetic over binary fields (pp474491)
P.R. Kaye
Shor's quantum algorithm for discrete logarithms applied
to elliptic curve groups forms the basis of a ``quantum attack'' of
elliptic curve cryptosystems. To implement this algorithm on a quantum
computer requires the efficient implementation of the elliptic curve
group operation. Such an implementation requires we be able to compute
inverses in the underlying field. In \cite{PZ03}, Proos and Zalka show
how to implement the extended Euclidean algorithm to compute inverses in
the prime field $\GF(p)$. They employ a number of optimizations to
achieve a running time of $O(n^2)$, and a spacerequirement of $O(n)$
qubits, where $n$ is the number of bits in the binary representation of
$p$ (there are some tradeoffs that they make, sacrificing a few extra
qubits to reduce runningtime). In practice, elliptic curve
cryptosystems often use curves over the binary field $\GF(2^m)$. In this
paper, I show how to implement the extended Euclidean algorithm for
polynomials to compute inverses in $\GF(2^m)$. Working under the
assumption that qubits will be an `expensive' resource in realistic
implementations, I optimize specifically to reduce the qubit space
requirement, while keeping the runningtime polynomial. The
implementation here differs from that in $\cite{PZ03}$ for $\GF(p)$, and
we are able to take advantage of some properties of the binary field
$\GF(2^m)$. I also optimize the overall qubit space requirement for
computing the group operation for elliptic curves over $\GF(2^m)$ by
decomposing the group operation to make it ``piecewise reversible''
(similar to what is done in \cite{PZ03} for curves over $\GF(p)$).
Physicallymotivated dynamical algorithms for the graph isomorphism
problem (pp492506)
S.Y. Shiau, R. Joynt and S.N.
Coppersmith
The graph isomorphism problem (GI) plays a central role
in the theory of computational complexity and has importance in physics
and chemistry as well \cite{kobler93,fortin96}. No polynomialtime
algorithm for solving GI is known. We investigate classical and quantum
physicsbased polynomialtime algorithms for solving the graph
isomorphism problem in which the graph structure is reflected in the
behavior of a dynamical system. We show that a classical dynamical
algorithm proposed by Gudkov and Nussinov \cite{gudkov02} as well as its
simplest quantum generalization fail to distinguish pairs of
nonisomorphic strongly regular graphs. However, by combining the
algorithm of Gudkov and Nussinov with a construction proposed by Rudolph
\cite{rudolph02} in which one examines a graph describing the dynamics
of two particles on the original graph, we find an algorithm that
successfully distinguishes all pairs of nonisomorphic strongly regular
graphs that we tested with up to 29 vertices.
Mini
tutorial:
A simple proof of the strong
subadditivity inequality (pp507513)
M.A. Nielsen and D. Petz
Arguably the deepest fact known about the
von~Neumann entropy, the strong subadditivity inequality is a potent
hammer in the quantum information theorist's toolkit. This short
tutorial describes a simple proof of strong subadditivity due to Petz
[Rep. on Math. Phys. \textbf{23} (1), 5765 (1986)]. It assumes only
knowledge of elementary linear algebra and quantum mechanics.
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