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**Research Articles:

**Fault-tolerant renormalization group decoder for
abelian topological codes**** **
(pp0721-0740)

Guillaume
Duclos-Cianci and David Poulin

We present a three-dimensional generalization of a renormalization
group decoding algorithm for topological codes with Abelian anyonic
excitations that we introduced for two dimensions in \cite{DP09a,DP10a}.
We also provide a complete detailed description of the structure of the
algorithm, which should be sufficient for anyone interested in
implementing it. This 3D implementation extends our previous 2D
algorithm by incorporating a failure probability of the syndrome
measurements, i.e., it enables fault-tolerant decoding. We report a
fault-tolerant storage threshold of $\sim1.9(4)\%$ for Kitaev's toric
code subject to a 3D bit-flip channel (i.e. including imperfect syndrome
measurements). This number is to be compared with the $2.9\%$ value
obtained via perfect matching \cite{H04a}. The 3D generalization
inherits many properties of the 2D algorithm, including a complexity
linear in the space-time volume of the memory, which can be parallelized
to logarithmic time.

**Dynamics of coupled cavity arrays embedded in a
non-Markovian bath**
(pp0741-0756)

Xinyu Zhao,
Jun Jing, J.Q. You, and Ting Yu

In this paper, the non-Markovian quantum dynamics of a coupled
$N$-cavity model is studied based on the quantum state diffusion (QSD)
approach. The time-local Di\'{o}si-Gisin-Strunz equation and the
corresponding exact master equation are derived for the model consisting
of a coupled cavity array. The resulting QSD equation serves as a
stochastic solution to a genuine $N$-partite continuous-variable (CV)
system. Several non-Markovian effects are studied in two interesting
examples -- two-cavity and three-cavity, under different boundary
conditions. We have shown that the environment-memory can facilitate the
cat-like state transfer from one cavity to another in the case of a
strongly non-Markovian environment.

**Unitary operation attack and the improvement on
probabilistic quantum key distribution **
(pp0757-0762)

Tzu-Han Lin,
Chun-Wei Yang, and Tzonelih Hwang

This study points out that a malicious communicant in Hwang et al.\textquoteright{}s
probabilistic quantum key distribution (PQKD) protocol can manipulate
the secret key without being detected by using the unitary operation
attack. Accordingly, the security requirements of a PQKD protocol,

i.e., unpredictability and fairness, cannot be satisfied in their
protocol. A possible solution is also provided to solve the problem.

**Quantum circuits for simple periodic functions **
(pp0763-0776)

Omar Gamel
and Daniel F.V. James

Periodic functions are of special importance in quantum computing,
particularly in applications of Shor's algorithm. We explore methods of
creating circuits for periodic functions to better understand their
properties. We introduce a method for constructing the circuit for a
simple monoperiodic function, that is one-to-one within a single period,
of a given period $p$. We conjecture that to create a simple periodic
function of period $p$, where $p$ is an $n$-bit number, one needs at
most $n$ Toffoli gates.

**Long distance entanglement generation through
coherent directed transport of**
**Neutral atoms in unmodulated optical lattices **
(pp0777-0789)

Morteza
Rafiee and Abolfazl Bayat

We introduce a fully coherent way for directed transport of
localized atoms in optical lattices by regularly performing phase shifts
on the lattice potential during the free evolution of the system. This
paves the way for realizing a possible cold atom quantum computer in
which entangling gates operate by bringing two individual atoms in the
proximity of each other and letting them to interact. The speed of our
protocol is determined by the tunneling amplitudes of the atoms and thus
is much faster than the speed of the dynamics resulted from
superexchange interaction in spin chains. Our scheme is robust against
possible imperfections and perhaps its main advantage is its simplicity
where all of its requirements have been already achieved in recent
experiments.

**Polynomial time quantum algorithms for certain
bivariate hidden polynomial problems **
(pp0790-0806)

Thomas
Decker, Peter Hoyer, Gabor Ivanyos, and Miklos Santha

We present a new method for solving the hidden polynomial graph
problem (HPGP) which is a special case of the hidden polynomial problem
(HPP). The new approach yields an efficient quantum algorithm for the
bivariate HPGP even when the input consists of several level set
superpositions, a more difficult version of the problem than the one
where the input is given by an oracle. For constant degree, the
algorithm is polylogarithmic in the size of the base field. We also
apply the results to give an efficient quantum algorithm for the oracle
version of the HPP for an interesting family of bivariate hidden
functions. This family includes diagonal quadratic forms and elliptic
curves.

**Performance and error analysis of Knill's
postselection scheme in a two-dimensional architecture **
(pp0807-0822)

Ching-Yi
Lai, Gerardo Paz, Martin Suchara, and Todd A. Brun

Knill demonstrated a fault-tolerant quantum computation scheme based
on concatenated error-detecting codes and postselection with a simulated
error threshold of $3\%$ over the depolarizing channel. We show how to
use Knill's postselection scheme in a practical two-dimensional quantum

architecture that we designed with the goal to optimize the error
correction properties, while satisfying important architectural
constraints. In our 2D architecture, one logical qubit is embedded in a
tile consisting of $5\times 5$ physical qubits. The movement of these
qubits is modeled as noisy SWAP gates and the only physical operations
that are allowed are local one- and two-qubit gates. We evaluate the
practical properties of our design, such as its error threshold, and
compare it to the concatenated Bacon-Shor code and the concatenated
Steane code. Assuming that all gates have the same error rates, we
obtain a threshold of $3.06\times 10^{-4}$ in a local adversarial
stochastic noise model, which is the highest known error threshold for
concatenated codes in 2D. We also present a Monte Carlo simulation of
the 2D architecture with depolarizing noise and we calculate a
pseudo-threshold of about $0.1\%$. With memory error rates one-tenth of
the worst gate error rates, the threshold for the adversarial noise
model, and the pseudo-threshold over depolarizing noise, are $4.06\times
10^{-4}$ and $0.2\%$, respectively. In a hypothetical technology where
memory error rates are negligible, these thresholds can be further
increased by shrinking the tiles into a $4\times 4$ layout.

**Unextendible mutually unbiased bases from Pauli
C\classes **
(pp0823-0844)

Prabha
Mandayam, Somshubhro Bandyopadhyay, Markus Grassl, and William K.
Wootters

We provide a construction of sets of $d/2+1$ mutually unbiased bases
(MUBs) in dimensions $d=4,8$ using maximal commuting classes of Pauli
operators. We show that these incomplete sets cannot be extended further
using the operators of the Pauli group. Moreover, specific examples of
sets of MUBs obtained using our construction are shown to be {\it
strongly unextendible}; that is, there does not exist another vector
that is unbiased with respect to the elements in the set. We conjecture
the existence of such unextendible sets in higher dimensions $d=2^{n}
(n>3) $ as well.} {Furthermore, we note an interesting connection
between these unextendible sets and state-independent proofs of the
Kochen-Specker Theorem for two-qubit systems. Our construction also
leads to a proof of the tightness of a $H_{2}$ entropic uncertainty
relation for any set of three MUBs constructed from Pauli classes in
$d=4$.

**Quantum key distribution: defeating collective
noise without reducing efficiency **
(pp0845-0856)

Song Lin,
Gong-De Guo, Fei Gao, and Xiao-Fen Liu

Decoherence-free subspace (DFS) is a valid solution to realize
quantum communication over a collective noise channel, and has been
widely studied. Generally speaking, replacing a qubit with a DFS state
will cause the reduction of communication efficiency. However, in this
letter, it is shown that some kinds of noises may not lower the
transmission rate of quantum key distribution. To illustrate it, we
propose two quantum key distribution protocols based on Bell states.
Here, two nonorthogonal and unbiased sets in a DFS are constructed by
linear combination of particles at different positions. Since $n-1$
classical bits are distributed by using $2n$ qubits in our protocols,
the transmission rate is close to that of noiseless BB84 protocol.
Furthermore, when considering the cost of transmitting classical bits,
the efficiencies of these protocols are even higher than that of BB84
protocol.

**On the spectral dependence of separable and
classical correlations in small quantum systems **
(pp0857-0887)

Gary
McConnell and David Jennings

We study the correlation structure of separable and classical states
in $2\times2$-~and~$2\times3$-dimensional quantum systems with fixed
spectra. Even for such simple systems the maximal correlation - as
measured by mutual information - over the set of unitarily accessible
separable states is highly non-trivial to compute; however for the
$2\times2$ case a particular class of spectra admits full analysis and
allows us to contrast classical states with more general separable
states. We analyse a particular entropic binary relation on the set of
spectra and prove for the qubit-qutrit case that this relation alone
picks out a unique classical maximum state for mutual information.
Moreover the $2\times3$ case is the largest system with such a property.

**A quantum circuit to find discrete logarithms on
ordinary binary elliptic curves in depth O(log^2n) **
(pp0888-0900)

Martin
Rotteler and Rainer Steinwandt

Improving over an earlier construction by Kaye and Zalka
\cite{KaZa04}, in \cite{MMCP09b} Maslov et al. describe an
implementation of Shor's algorithm, which can solve the discrete
logarithm problem on ordinary binary elliptic curves in quadratic depth
$\bigO(n^2)$. In this paper we show that discrete logarithms on such
curves can be found with a quantum circuit of depth $\bigO(\log^2n)$. As
technical tools we introduce quantum circuits for ${\mathbb
F}_{2^n}$-multiplication in depth $\bigO(\log n)$ and for ${\mathbb
F}_{2^n}$-inversion in depth $\bigO(\log^2 n)$.