Vol.16 No.3&4, March
1, 2016
Research Articles:
Local decoders for the 2D
and 4D toric code
(pp01810208)
Nikolas
P. Breuckmann, Kasper Duivenvoorden, Dominik Michels, and Barbara M.
Terhal
We analyze the performance of decoders
for the
2D
and 4D
toric
code which are local by construction. The
2D
decoder is a cellular automaton decoder formulated by Harrington
\cite{thesis:harrington}
which explicitly has a finite speed of communication and computation.
For a model of independent $X$
and $Z$
errors and faulty syndrome measurements with identical probability, we
report a threshold of $0.133\%$
for this Harrington decoder. We implement a decoder for the
4D
toric
code which is based on a decoder by Hastings
\cite{hastings}.
Incorporating a method for handling faulty syndromes we estimate a
threshold of $1.59\%$
for the same noise model as in the
2D
case. We compare the performance of this decoder with a decoder based on
a 4D
version of
Toom's
cellular automaton rule as well as the decoding method suggested by
Dennis {\em
et
al.}
\cite{DKLP}.
Quantum key distribution
with mismatched measurements over arbitrary channels
(pp02090241)
Walter
O. Krawec
In this paper, we derive keyrate
expressions for different quantum key distribution protocols. Our
keyrate equations utilize multiple channel statistics, including those
gathered from mismatched measurement bases  i.e., when Alice and Bob
choose incompatible bases. In particular, we will consider an
Extended
B92
and a twoway semiquantum protocol. For both these protocols, we
demonstrate that their tolerance to noise is higher than previously
thought  in fact, we will show the semiquantum protocol can actually
tolerate the same noise level as the fully quantum
BB84
protocol. Along the way, we will also consider an optimal
QKD
protocol for various quantum channels. Finally, all the keyrate
expressions which we derive in this paper are applicable to any
arbitrary, not necessarily symmetric, quantum channel.
Modified group
nonmembership is in promiseAWPP
relative
to group oracles
(pp02420250)
Tomoyuki
Morimae, Harumichi Nishimura, and
Francois Le Gall
It is known that the group
nonmembership problem is in
QMA
relative to any group oracle and in
${\rm SPP}\cap{\rm BQP}$
for solvable groups. We consider a modified version of the group
nonmembership problem where the order of the group is also given as an
additional input. We show that the problem is in (the promise
version of)
AWPP
relative to any group oracle. To show the result, we use the idea of
postselected quantum computing.
Optimizing the number of
gates in quantum search
(pp02510261)
Srinivasan
Arunachalam
and Ronald de Wolf
In its usual form, Grover's quantum
search algorithm uses $O(\sqrt{N})$
queries and $O(\sqrt{N} \log N)$
other elementary gates to find a solution in an
$N$bit
database. Grover in 2002 showed how to reduce the number of other
gates to $O(\sqrt{N}\log\log N)$
for the special case where the database has a unique solution, without
significantly increasing the number of queries. We show how to
reduce this further to $O(\sqrt{N}\log^{(r)}
N)$ gates for every constant~$r$,
and sufficiently large~$N$.
This means that, on average, the circuits between two queries barely
touch more than a constant number of the
$\log N$
qubits
on which the algorithm acts. For a very large
$N$
that is a power of~2, we can choose~$r$
such that the algorithm uses essentially the minimal number
$\frac{\pi}{4}\sqrt{N}$
of queries, and only $O(\sqrt{N}\log(\log^{\star}
N))$ other gates.
Further extensions of
Clifford circuits and their classical simulation complexities
(pp02620282)
Dax
E. Koh
Extended Clifford circuits straddle
the boundary between classical and quantum computational power. Whether
such circuits are efficiently classically
simulable
seems to depend delicately on the ingredients of the circuits. While
some combinations of ingredients lead to efficiently classically
simulable
circuits, other combinations, which might just be slightly different,
lead to circuits which are likely not. We extend the results of
Jozsa
and Van den Nest [Quant.
Info.
Comput. 14, 633 (2014)] by studying
two further extensions of Clifford circuits. First, we consider how the
classical simulation complexity changes when we allow for more general
measurements. Second, we investigate different notions of what it means
to `classically simulate' a quantum circuit. These further extensions
give us 24 new combinations of ingredients compared to
Jozsa
and Van den Nest, and we give a complete classification of their
classical simulation complexities. Our results provide more examples
where seemingly modest changes to the ingredients of Clifford circuits
lead to ``large" changes in the classical simulation complexities of the
circuits, and also include new examples of extended Clifford circuits
that exhibit ``quantum supremacy", in the sense that it is not possible
to efficiently classically sample from the output distributions of such
circuits, unless the polynomial hierarchy collapses.
On quantum additive
Gaussian noise channels
(pp02830302)
Martin
Idel and Robert Konig
We give necessary and sufficient
conditions for a Gaussian quantum channel to have a dilation
involving a passive, i.e., numberpreserving unitary. We then establish
a normal form of such channels: any passively
dilatable
channel is the result of applying passive
unitaries
to the input and output of a Gaussian additive channel. The latter
combine the state of the system with that of the environment by means of
a
multimode
beamsplitter.
Perfect state transfer on graphs with a potential
(pp03030327)
Mark
Kempton, Gabor Lippner, and ShingTung Yau
In this paper we study quantum state
transfer (also called quantum tunneling) on graphs when there is a
potential function on the vertex set. We present two main results.
First, we show that for paths of length greater than three, there is no
potential on the
vertices
of the path for which perfect state transfer between the endpoints can
occur. In particular, this answers a question raised by
Godsil
in Section 20 of~\cite{godsil}.
Second, we show that if a graph has two
vertices
that share a common neighborhood, then there is a potential on the
vertex set for which perfect state transfer will occur between those two
vertices. This gives numerous
examples where perfect state transfer does not occur without the
potential, but adding a potential makes perfect state transfer possible.
In addition, we investigate perfect state transfer on graph products,
which gives further examples where perfect state transfer can occur.
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