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Research Articles:*

**Integrated optical approach
to trapped ion quantum computation**
(pp0181-0202)

J.
Kim and C. Kim

Recent experimental progress in quantum information processing with
trapped ions have demonstrated most of the fundamental elements required
to realize a scalable quantum computer. The next set of challenges lie
in realization of a large number of qubits and the means to prepare,
manipulate and measure them, leading to error-protected qubits and fault
tolerant architectures. The integration of qubits necessarily require
integrated optical approach as most of these operations involve
interaction with photons. In this paper, we discuss integrated optics
technologies and concrete optical designs needed for the physical
realization of scalable quantum computer.

**Scalable, efficient ion-photon
coupling with phase Fresnel lenses for large-scale quantum computing**
(pp0203-0214)

E.W.
Streed, B.G. Norton, J.J. Chapman, and D. Kielpinski

Efficient ion-photon coupling is an important component for large-scale
ion-trap quantum computing. We propose that arrays of phase Fresnel
lenses (PFLs) are a favorable optical coupling technology to match with
multi-zone ion traps. Both are scalable technologies based on
conventional micro-fabrication techniques. The large numerical apertures
(NAs) possible with PFLs can reduce the readout time for ion qubits.
PFLs also provide good coherent ion-photon coupling by matching a large
fraction of an ion's emission pattern to a single optical propagation
mode (TEM$_{00}$). To this end we have optically characterized a large
numerical aperture phase Fresnel lens (NA=0.64) designed for use at
369.5 nm, the principal fluorescence detection transition for Yb$^+$
ions. A diffraction-limited spot $w_0=350\pm15$ nm ($1/e^2$ waist) with
mode quality $M^2= 1.08\pm0.05$ was measured with this PFL. From this we
estimate the minimum expected free space coherent ion-photon coupling to
be 0.64\%, which is twice the best previous experimental measurement
using a conventional multi-element lens. We also evaluate two techniques
for improving the entanglement fidelity between the ion state and photon
polarization with large numerical aperture lenses.

**Efficient quantum algorithm for
identifying hidden polynomials**
(pp0215-0230)

T.
Decker, J. Draisma, and P. Wocjan

We consider a natural generalization of an abelian Hidden Subgroup
Problem where the subgroups and their cosets correspond to graphs of
linear functions over a finite field $\F$ with $d$ elements. The hidden
functions of the generalized problem are not restricted to be linear but
can also be $m$-variate polynomial functions of total degree $n\geq 2$.
The problem of identifying hidden $m$-variate polynomials of degree less
or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer
since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a
constant success probability. In contrast, we present a quantum
algorithm that correctly identifies such hidden polynomials for all but
a finite number of values of $d$ with constant probability and that has
a running time that is only polylogarithmic in $d$.

**Graph embedding using quantum hitting
time**
(pp0231-0254)

D.
Emms, R. Wilson, and E. Hancock

In this paper, we explore analytically and experimentally a
quasi-quantum analogue of the hitting time of the continuous-time
quantum walk on a graph. For the classical random walk, the hitting time
has been shown to be robust to errors in edge weight structure and to
lead to spectral clustering algorithms with improved performance. Our
analysis shows that the quasi-quantum analogue of the hitting time of
the continuous-time quantum walk can be determined via integrals of the
Laplacian spectrum, calculated using Gauss-Laguerre quadrature. We
analyse the quantum hitting times with reference to their classical
counterpart. Specifically, we explore the graph embeddings that preserve
hitting time. Experimentally, we show that the quantum hitting times can
be used to emphasise cluster-structure.

**Communication complexities of
symmetric XOR functions**
(pp0255-0263)

Z.-Q.
Zhang and Y.-Y. Shi

We call $F:\{0, 1\}^n\times \{0, 1\}^n\to\{0, 1\}$ a symmetric XOR
function if for a function $S:\{0, 1, ..., n\}\to\{0, 1\}$, $F(x, y)=S(|x\oplus
y|)$, for any $x, y\in\{0, 1\}^n$, where $|x\oplus y|$ is the Hamming
weight of the bit-wise XOR of $x$ and $y$. We show that for any such
function, (a) the deterministic communication complexity is always $\Theta(n)$
except for four simple functions that have a constant complexity, and
(b) up to a polylog factor, both the error-bounded randomized complexity
and quantum communication with entanglement complexity are
$\Theta(r_0+r_1)$, where $r_0$ and $r_1$ are the minimum integers such
that $r_0, r_1\leq n/2$ and $S(k)=S(k+2)$ for all $k\in[r_0, n-r_1)$.

**Estimating Jones and HOMFLY
polynomials with one clean qubit**
(pp0264-0289)

S.P.
Jordan and P. Wocjan

The Jones and HOMFLY polynomials are link invariants with close
connections to quantum computing. It was recently shown that finding a
certain approximation to the Jones polynomial of the trace closure of a
braid at the fifth root of unity is a complete problem for the one clean
qubit complexity class\cite{Shor_Jordan}. This is the class of problems
solvable in polynomial time on a quantum computer acting on an initial
state in which one qubit is pure and the rest are maximally mixed. Here
we generalize this result by showing that one clean qubit computers can
efficiently approximate the Jones and single-variable HOMFLY polynomials
of the trace closure of a braid at \emph{any} root of unity.

**High fidelity universal set of
quantum gates using non-adiabatic rapid passage**
(pp0290-0316)

R.
Li, M. Hoover, and F. Gaitan

Numerical simulation results are presented which suggest that a class of
non-adiabatic rapid passage sweeps first realized experimentally in 1991
should be capable of implementing a universal set of quantum gates
\uniset\ that operate with high fidelity. The gates constituting \uniset\
are the Hadamard and NOT gates, together with variants of the phase,
$\pi /8$, and controlled-phase gates. The universality of \uniset\ is
established by showing that it can construct the universal set
consisting of Hadamard, phase, $\pi /8$, and controlled-NOT gates. Sweep
parameter values are provided which simulations indicate will produce
the different gates in \uniset , and for which the gate error
probability $P_{e}$ satisfies: (i)~$P_{e}<10^{-4}$ for the one-qubit
gates; and (ii)~$P_{e}<1.27\times 10^{-3}$ for the modified
controlled-phase gate. The sweeps in this class are non-composite and
generate controllable quantum interference effects that allow the gates
in \uniset\ to operate non-adiabatically while maintaining high
fidelity. These interference effects have been observed using NMR, and
it has previously been shown how these rapid passage sweeps can be
applied to atomic systems using electric fields. Here we show how these
sweeps can be applied to both superconducting charge and flux
qubit systems. The simulations suggest that the universal set of gates \uniset\
produced by these rapid passage sweeps shows promise as possible
elements of a fault-tolerant scheme for quantum computing.

**Non-Markovian decoherence dynamics of
entangled coherent states**
(pp0317-0335)

J.-H.
An, M. Feng, and W. M. Zhang

We microscopically model the decoherence dynamics of entangled coherent
states of two optical modes under the influence of vacuum fluctuation.
We derive an exact master equation with time-dependent coefficients
reflecting the memory effect of the environment, by using the
Feynman-Vernon influence functional theory in the coherent-state
representation. Under the Markov approximation, our master equation
recovers the widely used Lindblad equation in quantum optics. We then
investigate the non-Markovian entanglement dynamics of the two-mode
entangled coherent states under vacuum fluctuation. Compared with the
results in Markov limit, it shows that the non-Markovian effect enhances
the disentanglement to the initially entangled coherent state. Our
analysis also shows that the decoherence behaviors of the entangled
coherent states depend on the symmetrical properties of the entangled
coherent states as well as the couplings between the optical fields and
the environment.

**Classical and quantum tensor product
expanders**
(pp0336-0360)

M.B.
Hastings and A.W. Harrow

We introduce the concept of quantum tensor product expanders. These
generalize the concept of quantum expanders, which are quantum maps that
are efficient randomizers and use only a small number of Kraus
operators. Quantum tensor product expanders act on several copies of a
given system, where the Kraus operators are tensor products of the Kraus
operator on a single system. We begin with the classical case, and show
that a classical two-copy expander can be used to produce a quantum
expander. We then discuss the quantum case and give applications to the
Solovay-Kitaev problem. We give probabilistic constructions in both
classical and quantum cases, giving tight bounds on the expectation
value of the largest nontrivial eigenvalue in the quantum case.

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