Vol.15
No.7&8, May
1, 2015
Research Articles:
JordanWigner formalism for
arbitrary 2input 2output matchgates and their classical simulation
(pp05410556)
Richard
Jozsa, Akimasa Miyake, and Sergii Strelchuk
In Valiant's matchgate theory, 2input 2output matchgates are $4\times
4$ matrices that satisfy ten socalled matchgate identities. We prove
that the set of all such matchgates (including nonunitary and
noninvertible ones) coincides with the topological closure of the set
of all matrices obtained as exponentials of linear combinations of the
2qubit JordanWigner (JW) operators and their quadratic products,
extending a previous result of Knill. In Valiant's theory, outputs of
matchgate circuits can be classically computed in polytime. Via the JW
formalism, Terhal \& DiVincenzo and Knill established a relation of a
unitary class of these circuits to the efficient simulation of
noninteracting fermions. We describe how the JW formalism may be used
to give an efficient simulation for all cases in Valiant's simulation
theorem, which in particular includes the case of noninteracting
fermions generalised to allow arbitrary 1qubit gates on the first line
at any stage in the circuit. Finally we give an exposition of how these
simulation results can be alternatively understood from some basic Lie
algebra theory, in terms of a formalism introduced by Somma et al.
A note on the quantum collision
and set equality problems
(pp05570567)
Mark
Zhandry
A collision for a function $f$ is two distinct inputs $x_1\neq
x_2$ such that $f$ outputs the same value on both inputs:
$f(x_1)=f(x_2)$. The quantum query complexity of finding collisions has
been shown~\cite{BHT1997,AS2004, Ambainis05, Kutin05} in some settings
to be $\Theta(N^{1/3})$; however, these results do not apply to random
functions. The issues are twofold. First, the $\Omega(N^{1/3})$ lower
bound only applies when the domain is no larger than the codomain,
which precludes many of the cryptographically interesting applications.
Second, most of the results in the literature only apply to $r$to1
functions, which are quite different from random functions.
Understanding the collision problem for random functions is of
great importance to cryptography, and we seek to fill the gaps of
knowledge for this problem. To that end, we prove that, as expected, a
quantum query complexity of $\Theta(N^{1/3})$ holds for all interesting
domain and codomain sizes. Our proofs are simple, and combine existing
techniques with several novel tricks to obtain the desired results.
Using our techniques, we also give an optimal $\Omega(M^{1/3})$ lower
bound for the set equality problem. This lower bound can be used to
improve the relationship between classical randomized query complexity
and quantum query complexity for socalled permutationsymmetric
functions.
Improving the quality of noisy spatial quantum channels
(pp05680581)
Ning Tang,
ZiLong Fan, and HaoSheng Zeng
We show, for the nonMarkovian or timedependent Markovian model of
noise, by breaking the noisy spatial quantum channel (SQC) into a series
of periodically arranged subcomponents, that the quality of information
transmission described by the purity, fidelity and concurrence of the
output states can be improved. The physical mechanism and possible
implementation of the idea have been discussed.
Quantum state transfer in
disordered spin chains: How much engineering is reasonable?
(pp05820600)
Analia
Zwick, Gonzalo A. Alvarez, Joachim Stolze, and Omar Osenda
The transmission of quantum states through spin chains is an important
element in the implementation of quantum information technologies. Speed
and fidelity of transfer are the main objectives which have to be
achieved by the devices even in the presence of imperfections which are
unavoidable in any manufacturing process. To reach these goals, several
kinds of spin chains have been suggested, which differ in the degree of
finetuning, or engineering, of the system parameters. In this work we
present a systematic study of two important classes of such chains. In
one class only the spin couplings at the ends of the chain have to be
adjusted to a value different from the bulk coupling constant, while in
the other class every coupling has to have a specific value. We
demonstrate that configurations from the two different classes may
perform similarly when subjected to the same kind of disorder in spite
of the large difference in the engineering effort necessary to prepare
the system. We identify the system features responsible for these
similarities and we perform a detailed study of the transfer fidelity as
a function of chain length and disorder strength, yielding empirical
scaling laws for the fidelity which are similar for all kinds of chain
and all disorder models. These results are helpful in identifying the
optimal spin chain for a given quantum information transfer task. In
particular, they help in judging whether it is worthwhile to engineer
all couplings in the chain as compared to adjusting only the boundary
couplings.
Momentum switches 601
(pp06010621)
Andrew M.
Childs, David Gosset, Daniel Nagaj, Mouktik Raha, and Zak Webb
Certain continuoustime quantum walks can be viewed as scattering
processes. These processes can perform quantum computations, but it is
challenging to design graphs with desired scattering behavior. In this
paper, we study and construct momentum switches, graphs that route
particles depending on their momenta. We also give an example where
there is no exact momentum switch, although we construct an arbitrarily
good approximation.
There is entanglement in the
primes
(pp06220659)
Jose I.
Latorre and German Sierra
Large series of prime numbers can be superposed on a single quantum
register and then analyzed in full parallelism. The construction of this
Prime state is efficient, as it hinges on the use of a quantum version
of any efficient primality test. We show that the Prime state turns out
to be very entangled as shown by the scaling properties of purity, Renyi
entropy and von Neumann entropy. An analytical approximation to these
measures of entanglement can be obtained from the detailed analysis of
the entanglement spectrum of the Prime state, which in turn produces new
insights in the HardyLittlewood conjecture for the pairwise
distribution of primes. The extension of these ideas to a Twin Prime
state shows that this new state is even more entangled than the Prime
state, obeying majorization relations. We further discuss the
construction of quantum states that encompass relevant series of numbers
and opens the possibility of applying quantum computation to Arithmetics
in novel ways.
Quantum relay cooperative
communication via spacetime transmission
(pp06600676)
Jinjing
Shi, Ronghua Shi, Xiaoqi Peng, Ying Guo, and Moon Ho Lee
Two novel quantum relay cooperative communication schemes $1S2R1D$ and
$2S2R2D$ are proposed by applying the wireless communication technique
in the quantum system. The orthogonal quantum signals in consecutive
quantum sequences are prepared to avoid mutual interference and they are
firstly transmitted to the quantum relay node and then to the
destination node with spacetime transmission. Moreover, quantum
operations and transformations are implemented on the relay node and the
destination node to maximize the correct information transmission. The
fidelity between original and final received signals, the security, the
quantum bit error rate and the lower bound on the capacity of quantum
relay channels are analyzed. Especially it provides a valuable
application prospect in long distance reliable quantum communications.
Wlike states are not necessary
for totally correct quantum anonymous leader election
(pp06770684)
Alexander
Norton
I show that $W$like entangled quantum states are not a necessary
quantum resource for totally correct anonymous leader election
protocols. This is proven by defining a symmetric quantum state that is
$n$partite SLOCC inequivalent to the $W$ state, and then constructing a
totally correct anonymous leader election protocol using this state.
This result, which contradicts the previous necessity result of D'Hondt
and Panangaden, furthers our understanding of how nonlocal quantum
states can be used as a resource for distributed computation.
Solution to timeenergy costs
of quantum channels
(pp06850693)
ChiHang F.
Fung, H. F. Chau, ChiKwong Li, and NungSing Sze
We derive a formula for the timeenergy costs of general quantum
channels proposed in [Phys. Rev. A {\bf 88}, 012307 (2013)]. This
formula allows us to numerically find the timeenergy cost of any
quantum channel using positive semidefinite programming. We also derive
a lower bound to the timeenergy cost for any channels and the exact the
timeenergy cost for a class of channels which includes the qudit
depolarizing channels and projector channels as special cases.
Is absolute separability
determined by the partial transpose?
(pp06940720)
Srinivasan
Arunachalam, Nathaniel Johnston, and Vincent Russo
The absolute separability problem asks for a characterization of the
quantum states $\rho \in M_m\otimes M_n$ with the property that $U\rho
U^\dagger$ is separable for all unitary matrices $U$. We investigate
whether or not it is the case that $\rho$ is absolutely separable if and
only if $U\rho U^\dagger$ has positive partial transpose for all unitary
matrices $U$. In particular, we develop an easytouse method for
showing that an entanglement witness or positive map is unable to detect
entanglement in any such state, and we apply our method to many
wellknown separability criteria, including the range criterion, the
realignment criterion, the Choi map and its generalizations, and the
BreuerHall map. We also show that these two properties coincide for
the family of isotropic states, and several eigenvalue results for
entanglement witnesses are proved along the way that are of independent
interest.
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