Research Articles:
Computing quopit Clifford
circuit amplitudes by the sum-over-paths technique
(pp1081-1095)
Dax
E. Koh, Mark D. Penney, and Robert W. Spekkens
By the
Gottesman-Knill
Theorem, the outcome probabilities of Clifford circuits can be computed
efficiently. We present an alternative proof of this result for
quopit
Clifford circuits (i.e., Clifford circuits on collections of
$p$-level
systems, where $p$
is an odd prime) using Feynman's sum-over-paths technique, which allows
the amplitudes of arbitrary quantum circuits to be expressed in terms of
a weighted sum over computational paths. For a general quantum circuit,
the sum over paths contains an exponential number of terms, and no
efficient classical algorithm is known that can compute the sum. For
quopit
Clifford circuits, however, we show that the sum over paths takes a
special form: it can be expressed as a product of
Weil
sums with quadratic polynomials, which can be computed efficiently. This
provides a method for computing the outcome probabilities and amplitudes
of such circuits efficiently, and is an application of the
circuit-polynomial correspondence which relates quantum circuits to
low-degree polynomials.
Efficient implementation
of quantum circuits with limited qubit interactions
(pp1096-1104)
Stephen
Brierley
The quantum circuit model
allows gates between \emph{any}
pair of
qubits
yet physical
instantiations
allow only limited interactions. We address this problem by providing an
interaction graph together with an efficient method for compiling
quantum circuits so that gates are applied only locally. The graph
requires each
qubit
to interact with $4$
other
qubits and yet the time-overhead
for implementing any $n$-qubit
quantum circuit is $4\log n$.
Building a network of quantum computing nodes according to this graph
enables the network to emulate a single monolithic device with minimal
overhead.
On quantum tensor product codes
(pp1105-1122)
Jihao
Fan, Yonghui Li, Min-Hsiu Hsieh, and Hanwu Chen
We present a general
framework for the construction of quantum tensor product codes (QTPC).
In a classical tensor product code (TPC),
its parity check matrix is constructed via the tensor product of parity
check matrices of the two component codes. We show that by adding
some constraints on the component codes,
several classes of dual-containing
TPCs
can be obtained. By selecting different types of component codes, the
proposed method enables the construction of a large family of
QTPCs
and they can provide a wide variety of quantum error control abilities.
In particular, if one of the component codes is selected as a
burst-error-correction code, then
QTPCs
have quantum multiple-burst-error-correction abilities, provided these
bursts fall in distinct
subblocks.
Compared with concatenated quantum codes (CQC),
the component code selections of
QTPCs
are much more flexible than those of
CQCs
since only one of the component codes of
QTPCs
needs to satisfy the dual-containing restriction. We show that it
is possible to construct
QTPCs
with parameters better than other classes of quantum
error-correction codes (QECC),
e.g.,
CQCs and quantum
BCH
codes. Many
QTPCs
are obtained with parameters better than previously known quantum codes
available in the literature. Several classes of
QTPCs
that can correct multiple quantum bursts of errors are constructed based
on reversible cyclic codes and maximum-distance-separable (MDS)
codes.
Efficient rate-adaptive
reconciliation for CV-QKD protocol
(pp1123-1134)
Xiangyu
Wang, Yichen Zhang, Zhengyu Li, Bingjie Xu, Song Yu, and Hong Guo
Information reconciliation
protocol has a significant effect on the secret key rate and maximal
transmission distance of continuous-variable quantum key distribution
(CV-QKD) systems. We propose an efficient rate-adaptive reconciliation
protocol suitable for practical CV-QKD systems with time-varying quantum
channel. This protocol changes the code rate of multi-edge
type low density parity check codes, by puncturing (increasing the code
rate) and shortening (decreasing the code rate) techniques, to enlarge
the correctable signal-to-noise ratios regime, thus improves the overall
reconciliation efficiency comparing to the original fixed rate
reconciliation protocol. We verify our rate-adaptive reconciliation
protocol with three typical code rate, i.e., 0.1, 0.05 and 0.02, the
reconciliation efficiency keep around 93.5\%,
95.4\%
and 96.4\%
for different signal-to-noise ratios, which shows the potential of
implementing high-performance CV-QKD
systems using single code rate matrix.
Topological proofs of
contextuality in quantum mechanics
(pp1135-1166)
Cihan
Okay, Sam Roberts, Stephen D. Bartlett, and Robert Raussendorf
We provide a
cohomological
framework for
contextuality
of quantum mechanics that is suited to describing
contextuality
as a resource in measurement-based quantum computation. This framework
applies to the parity proofs first discussed by
Mermin,
as well as a different type of
contextuality
proofs based on symmetry transformations. The topological arguments
presented can be used in the state-dependent and the state-independent
case.
Robustness of QMA against
witness noise
(pp1167-1190)
Friederike
Anna Dziemba
Using the tool of concatenated
stabilizer coding, we prove that the complexity class
$\QMA$
remains unchanged even if every witness
qubit
is disturbed by constant noise. This result may not only be relevant for
physical implementations of verifying protocols but also attacking the
relationship between the complexity classes
$\QMA$,
$\QCMA$
and $\BQP$,
which can be reformulated in this unified framework of a verifying
protocol receiving a disturbed witness. While
$\QCMA$
and $\BQP$
are described by fully
dephasing
and depolarizing channels on the witness
qubits,
respectively, our result proves $\QMA$
to be robust against 27%
dephasing and
18%
depolarizing noise.
Small Majorana fermion
codes
(pp1191-1205)
Mathew
B. Hastings
We consider
Majorana
fermion
stabilizer codes with small number of modes and distance. We give
an upper bound on the number of logical
qubits
for distance $4$
codes, and we construct
Majorana
fermion
codes similar to the classical Hamming code that saturate this bound.
We perform numerical studies and find other distance
$4$
and $6$
codes that we conjecture have the largest possible number of logical
qubits
for the given number of physical
Majorana
modes. Some of these codes have more logical
qubits
than any
Majorana
fermion
code derived from a
qubit
stabilizer code.
A note on cohering
power and de-cohering power
(pp1206-1220)
Kaifeng
Bu and Chunhe Xiong
Cohering power and de-cohering
power have recently been proposed to quantify the ability of a quantum
operation to produce and erase coherence respectively. In this paper, we
investigate the properties of cohering power and de-cohering power.
First, we prove the equivalence between two different kinds of cohering
power for any quantum operation on single qubit
systems, which implies that $l_1$
norm of coherence is monotone under Maximally incoherent operation (MIO)
and Dephasing-covariant operation (DIO) in 2-dimensional space. In
higher dimensions, however, we show that the monotonicity under MIO or
DIO does not hold. Besides, we compare the set of quantum
operations with zero cohering power with Maximally incoherent operation
(MIO) and Incoherent operation (IO). Moreover, two different types of
de-cohering power are defined and we find that they are not equal
in single qubit systems. Finally, we make a comparison between cohering
power and de-cohering power for single qubit unitary operations and show
that cohering power is always larger than de-cohering power.