Research Articles:

**Noise thresholds for the [4,2,2]-concatenated
toric code** (pp1261-1281)

Ben Criger and
Barbara Terhal

We analyze the properties of a 2D topological code derived by
concatenating the \cfour code with the toric/surface code, or
alternatively by removing check operators from the 2D square-octagon or
4.8.8 color code. We show that the resulting code has a circuit-based
noise threshold of $\sim 0.41\%$ (compared to $\sim 0.6\%$ for the toric
code in a similar scenario), which is higher than any known 2D color
code. We believe that the construction may be of interest for hardware
in which one wants to use both long-range two-qubit gates as well as
short-range gates between small clusters of qubits.

**A note on coherence power of n-dimensional unitary
operators** (pp1282-1294)

Maria
Garcia-Diaz, Dario Egloff, and Martin B. Plenio

The coherence power of a quantum channel, that is, its maximum
ability to increase the coherence of input states, is a fundamental
concept within the framework of the resource theory of coherence. In
this note we discuss various possible definitions of coherence power and
coherence rate and their basic properties. Then we prove that the
coherence power of a unitary operator acting on a qubit, computed with
respect to the $l_1$-coherence measure, can be calculated by maximizing
its coherence gain over pure incoherent states. We proceed to show that
this result fails in the general case, that is, the maximal coherence
gain is found when acting on a state with non-vanishing coherence in the
case of the l1-coherence and dimension $N>2$, the relative entropy of
coherence and the geometric measure of coherence.

**Corrected quantum walk for optimal Hamiltonian
simulation** (pp1295-1317)

Dominic Berry
and Leonardo Novo

We describe a method to simulate Hamiltonian evolution on a quantum
computer by repeatedly using a superposition of steps of a quantum walk,
then applying a correction to the weightings for the numbers of steps of
the quantum walk. This correction enables us to obtain complexity which
is the same as the lower bound up to double-logarithmic factors for all
parameter regimes. The scaling of the query complexity is $O\left( \tau
\frac{\log\log\tau}{\log\log\log\tau} + \log(1/\epsilon) \right)$ where
$\tau := t\|H\|_{\max}d$, for $\epsilon$ the allowable error, $t$ the
time, $\|H\|_{\max}$ the max-norm of the Hamiltonian, and $d$ the
sparseness. This technique should also be useful for improving the
scaling of the Taylor series approach to simulation, which is relevant
to applications such as quantum chemistry.

**Quantum Latin squares and unitary error bases** (pp1318-1332)

Benjamin Musto
and Jamie Vicary

In this paper we introduce \textit{quantum Latin squares},
combinatorial quantum objects which generalize classical Latin squares,
and investigate their applications in quantum computer science. Our main
results are on applications to \textit{unitary error bases} (UEBs),
basic structures in quantum information which lie at the heart of
procedures such as teleportation, dense coding and error correction. We
present a new method for constructing a UEB from a quantum Latin square
equipped with extra data. Developing construction techniques for UEBs
has been a major activity in quantum computation, with three primary
methods proposed: \textit{shift-and-multiply}, \textit{Hadamard}, and \textit{group-theoretic}.
We show that our new approach simultaneously generalizes the
shift-and-multiply and Hadamard methods. Furthermore, we explicitly
construct a UEB using our technique which we prove cannot be obtained
from any of these existing methods.

**Generalized Dicke states** (pp1333-1348)

Stephan
Hartmann

Quantum master equations are an important tool in quantum optics and
quantum information theory. For systems comprising a small to medium
number of atoms (or qubits), the non-truncated equations are usually
solved numerically. In this paper, we present a group-theoretical
superoperator method that helps solving these equations. To do so, we
exploit the $SU(4)$-symmetry of the respective Lindblad operator and
construct basis states that generalize the well-known Dicke states. This
allows us to solve various problems analytically and to considerably
reduce the complexity of problems that can only be solved numerically.
Finally, we present three examples that illustrate the proposed method.

**Extension of remotely creatable region via local
unitary transformation on receiver side** (pp1349-1364)

G. Bochkin and
A. Zenchuk

We show that the length of the effective remote state creation via
the homogeneous spin-1/2 chain can be increased more than three times
using the local unitary transformation of the so-called extended
receiver (i.e., receiver joined with the nearest node(s)). This
transformation is *most* effective in the models with all-node
interactions. We consider an example of communication lines with the
two-qubit sender, one-qubit receiver and two-qubit extended receiver.

**Stability of two interacting entangled spins
interacting with a thermal environment** (pp1365-1378)

S. Dehdashti,
M. B. Harouni, Z. Harsij, J. Shen, H. Wang, Z. Xu, B. Mirza, and H. Chen

We study the entanglement dynamics of two entangled spins coupled
with a common environment consisting of a large number of harmonic
oscillators. Specifically, we study the impacts of both interaction and
temperature of the environment on the dynamic quantum correlation,
namely, entanglement and quantum discord of two spins via concurrence
and global quantum discord criteria. In the present system, we show that
the interaction between the spin sub-systems and the common environment
causes environmental states to approach a composition of even and odd
coherent states, which have different phases, and which are entangled
with the spin states. Moreover, using the thermofield approach, we
demonstrate quantum correlation stabilization as a result of increasing
environmental interaction as well as increasing temperature.

**The Clifford group forms a unitary 3-design** (pp1379-1400)

Zak Webb

Unitary $k$-designs are finite ensembles of unitary matrices
that approximate the Haar distribution over unitary matrices. Several
ensembles are known to be 2-designs, including the uniform distribution
over the Clifford group, but no family of ensembles was previously known
to form a 3-design. We prove that the Clifford group is a 3-design,
showing that it is a better approximation to Haar-random unitaries than
previously expected. Our proof strategy works for any distribution of
unitaries satisfying a property we call Pauli 2-mixing and proceeds
without the use of heavy mathematical machinery. We also show that the
Clifford group does not form a 4-design, thus characterizing how well
random Clifford elements approximate Haar-random unitaries.
Additionally, we show that the generalized Clifford group for qudits is
not a 3-design unless the dimension of the qudit is a power of 2.