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*Research Articles:

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*Near-linear constructions of exact unitary 2-designs (pp0721-0756)

Richard Cleve,
Debbie Leung, Li Liu, and Chunhao Wang

A unitary 2-design can be viewed as a quantum analogue of a
2-universal hash function: it is indistinguishable from a truly random
unitary by any procedure that queries it twice. We show that exact
unitary 2-designs on $n$ qubits can be implemented by quantum circuits
consisting of $\widetilde{O}(n)$ elementary gates in logarithmic depth.
This is essentially a quadratic improvement in size (and in width times
depth) over all previous implementations that are exact or approximate
(for sufficiently strong approximations).

**The Unruh effect interpreted as a quantum noise
channel** (pp0757-0770)

S. Omkar, R.
Srikanth, Subhashish Banerjee, and Ashutosh Kumar Alok

We make use of the tools of quantum information theory to shed light
on the Unruh effect. A modal qubit appears as if subjected to quantum
noise that degrades quantum information, as observed in the accelerated
reference frame. The Unruh effect experienced by a mode of a free Dirac
field, as seen by a relativistically accelerated observer, is treated as
a noise channel, which we term the Unruh channel. We characterize this
channel by providing its operator-sum representation, and study various
facets of quantum correlations, such as, Bell inequality violations,
entanglement, teleportation and measurement-induced decoherence under
the effect. We compare and contrast this channel from conventional noise
due to environmental decoherence. We show that the Unruh effect produces
an amplitude-damping-like channel, associated with zero temperature,
even though the Unruh effect is associated with a non-zero temperature.
Asymptotically, the Bloch sphere subjected to the channel does not
converge to a point, as would be expected by fluctuation-dissipation
arguments, but contracts by a finite factor. We construct for the Unruh
effect the inverse channel, a non-completely-positive map, that formally
reverses the effect, and offer some physical interpretation.

**Classification of transversal gates in qubit
stabilizer codes** (pp0771-0802)

Jonas T.
Anderson and Tomas Jochym-O'Connor

This work classifies the set of diagonal gates that can implement a
single or two-qubit transversal logical gate for qubit stabilizer codes.
We show that individual physical diagonal gates on the underlying qubits
that compose the code are restricted to have entries of the form~$e^{i
\pi c/2^k}$ along their diagonal, resulting in a similarly restricted
class of logical gates that can be implemented in this manner. As such,
we show that all diagonal logical gates that can be implemented
transversally by individual physical diagonal gates must belong to the
Clifford hierarchy. Moreover, we show that for a given stabilizer code,
the two-qubit diagonal transversal gates must belong to the same level
of Clifford hierarchy as the single-qubit diagonal transversal gates
available for the given code. We use this result to prove a conjecture
about arbitrary transversal gates made by Zeng et al in 2007.

**Phase estimation using an approximate eigenstate** (pp0803-0812)

Avatar Tulsi

A basic building block of many quantum algorithms is the Phase
Estimation algorithm (PEA). It finds an eigenphase $\phi$ of a unitary
operator using a copy of the corresponding eigenstate $|\phi\rangle$.
Suppose, in place of $|\phi\rangle$, we have a copy of an approximate
eigenstate $|\psi\rangle$ whose component in $|\phi\rangle$ is at least
$\sqrt{2/3}$. Then the PEA fails with a constant probability. Using
multiple copies of $|\psi\rangle$, this probability can be made to
decrease exponentially with the number of copies. Here we show that a
single copy is sufficient to find $\phi$ if we can selectively invert
the $|\psi\rangle$ state. As an application, we consider the eigenpath
traversal problem (ETP) where the goal is to travel a path of
non-degenerate eigenstates of $n$ different operators. The fastest
algorithm for ETP is due to Boixo, Knill and Somma (BKS) which needs
$\Theta(\ln n)$ copies of the eigenstates. Using our method, the BKS
algorithm can work with just a single copy but its running time $\mathcal{Q}$
increases to $O(\mathcal{Q}\ln^{2}\mathcal{Q})$. This tradeoff is
beneficial if the spatial resources are more constrained than the
temporal resources.

**An improved asymptotic key rate bound for a
mediated semi-quantum key distribution protocol** (pp0813-0834)

Walter O.
Krawec

Semi-quantum key distribution (SQKD) protocols allow for the
establishment of a secret key between two users Alice and Bob, when one
of the two users (typically Bob) is limited or ``classical'' in nature.
Recently it was shown that protocols exists when both parties are
limited/classical in nature if they utilize the services of a quantum
server. These protocols are called mediated SQKD protocols. This server,
however, is untrusted and, in fact, adversarial. In this paper, we
reconsider a mediated SQKD protocol and derive a new proof of
unconditional security for it. In particular, we derive a new lower
bound on its key rate in the asymptotic scenario. Furthermore, we show
this new lower bound is an improvement over prior work, thus showing
that the protocol in question can tolerate higher rates of error than
previously thought.

**Renormalization of quantum deficit and monogamy
relation in the Heisenberg XXZ model** (pp0835-0844)

Meng Qin, Xin
Zhang**,** and Zhong-Zhou Ren

In this study, the dynamical behavior of quantum deficit and
monogamy relation in the Heisenberg XXZ model is investigated by
implementing quantum renormalization group theory. The results
demonstrate that the quantum deficit can be used to capture the quantum
phase transitions point and show scaling behavior with the spin chain
size increasing. It was also found that the critical exponent has no
change when varying measure from entanglement to quantum correlation.
The monogamy relation is influenced by the steps of quantum
renormalization group and the ways of splitting the block states.
Furthermore, the monogamy relation of generalized $W$ state also is
given by means of quantum deficit.

**Optimal bounds on functions of quantum states
under quantum channels** (pp0845-0861)

Chi-Kwong Li,
Diane Christine Pelejo, and Kuo-Zhong Wang

Let $\rho_1, \rho_2$ be quantum states and $(\rho_1,\rho_2) \mapsto
D(\rho_1, \rho_2)$ be a scalar function such as the trace distance, the
fidelity, and the relative entropy, etc. We determine optimal bounds for
$D(\rho_1, \Phi(\rho_2))$ for $\Phi \in \mathcal{S}$ for different class
of functions $D(\cdot, \cdot)$, where $\mathcal{S}$ is the set of
unitary quantum channels, the set of mixed unitary channels, the set of
unital quantum channels, and the set of all quantum channels.

**Improved quantum ternary arithmetic** (pp0862-0884)

Alex Bocharov,
Shawn X. Cui, Martin Roetteler, and Krysta M. Svore

Qutrit (or ternary) structures arise naturally in many quantum
systems, notably in certain non-abelian anyon systems. We present
efficient circuits for ternary reversible and quantum arithmetics. Our
main result is the derivation of circuits for two families of ternary
quantum adders. The main distinction from the binary adders is a richer
ternary carry which leads potentially to higher resource counts in
universal ternary bases. Our ternary ripple adder circuit has a circuit
depth of $O(n)$ and uses only $1$ ancilla, making it more efficient in
both, circuit depth and width, when compared with previous
constructions. Our ternary carry lookahead circuit has a circuit depth
of only $O(\log\,n)$, while using $O(n)$ ancillas. Our approach works on
two levels of abstraction: at the first level, descriptions of
arithmetic circuits are given in terms of gates sequences that use
various types of non-Clifford reflections. At the second level, we break
down these reflections further by deriving them either from the two-qutrit
Clifford gates and the non-Clifford gate $C(X): \ket{i,j}\mapsto \ket{i,
j + \delta_{i,2} \; {\rm mod} \; 3}$ or from the two-qutrit Clifford
gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 \pi \, i/9},1,e^{2
\pi \, i/9})$. The two choices of elementary gate sets correspond to two
possible mappings onto two different prospective quantum computing
architectures which we call the metaplectic and the supermetaplectic
basis, respectively. Finally, we develop a method to factor diagonal
unitaries using multi-variate polynomials over the ternary finite field
which allows to characterize classes of gates that can be implemented
exactly over the supermetaplectic basis.

**Phase diagrams of one-, two-, and
three-dimensional quantum spin systems** (pp0885-0899)

Briiissuurs
Braiorr-Orrs, Michael Weyrauch, and Mykhailo V. Rakov

We study the bipartite entanglement per bond to determine
characteristic features of the phase diagram of various quantum spin
models in different spatial dimensions. The bipartite entanglement is
obtained from a tensor network representation of the ground state
wave-function. Three spin-1/2 models (Ising, XY, XXZ, all in a
transverse field) are investigated. Infinite imaginary-time evolution (iTEBD
in 1D, `simple update' in 2D and 3D) is used to determine the ground
states of these models. The phase structure of the models is discussed
for all three dimensions.