Vol.7
No.5&6
July 1, 2007
Research Articles:
Matrix product state representations (pp401430)
D.
PerezGarcia, F. Verstraete, M.M. Wolf, and J.I. Cirac
This work gives a detailed investigation of matrix
product state (MPS) representations for pure multipartite quantum
states. We determine the freedom in representations with and without
translation symmetry, derive respective canonical forms and provide
efficient methods for obtaining them. Results on frustration free
Hamiltonians and the generation of MPS are extended, and the use of the
MPSrepresentation for classical simulations of quantum systems is
discussed.
Security of quantum key distribution
using weak coherent states with nonrandom phases (pp431458)
H.K.
Lo and J. Preskill
We prove the security of the BennettBrassard (BB84)
quantum key distribution protocol in the case where the key information
is encoded in the relative phase of a coherentstate reference pulse and
a weak coherentstate signal pulse, as in some practical implementations
of the protocol. In contrast to previous work, our proof applies even if
the eavesdropper knows the phase of the reference pulse, provided that
this phase is not modulated by the source, and even if the reference
pulse is bright. The proof also applies to the case where the key is
encoded in the photon polarization of a weak coherentstate pulse with a
known phase, but only if the phases of the four BB84 signal states are
judiciously chosen. The achievable key generation rate scales
quadratically with the transmission in the channel, just as for BB84
with phaserandomized weak coherentstate signals (when decoy states are
not used). For the case where the phase of the reference pulse is
strongly modulated by the source, we exhibit an explicit attack that
allows the eavesdropper to learn every key bit in a parameter regime
where a protocol using phaserandomized signals is provably secure.
Evolution from entanglement to
decoherence (pp459468)
T.
Yu and J.H. Eberly
We examine a class of bipartite mixed states which we
call X states and report several analytic results related to the
occurrence of earlystage decoherence or socalled entanglement sudden
death (ESD) under time evolution in the presence of common types of
environmental noise. Avoidance of sudden death by application of purely
local operations is shown to be feasible in some cases.
Controllable Subsystems of Quantum
Dynamical Systems (pp469478)
M.
Zhang, H.Y. Dai, G.H. Dong, H.W. Xie, and D.W. Hu
This paper proposes the concept of subsystem
controllability for both closed and open quantum dynamical systems, and
the existence conditions of controllable subsystems are derived
accordingly. It is demonstrated that a controllable subsystem may exist
for a closed quantum dynamical system even if the overall system is not
controllable. Furthermore, we also demonstrate that, for a class of
decoherencefree subsystems of Markovian open quantum dynamical
systems, the subsystems are controllable under openloop coherent
control if the initial states of the open systems are factorized. To a
certain extent, this paper gives an alternative control theoretical
interpretation on why decoherencefree subsystems are useful for quantum
computation.
Quantum automata, braid group and
link polynomials (pp479503)
S.
Garnerone, A. Marzuoli, and M. Rasetti
The spinnetwork quantum simulator model, which
essentially encodes
the (quantum deformed) $SU(2)$ RacahWigner tensor algebra, is
particularly suitable to address problems arising in low dimensional
topology and group theory. In this combinatorial framework we implement
families of finitestates and discretetime quantum automata capable
of accepting the language generated by the braid group, and whose
transition amplitudes are colored Jones polynomials. The automaton
calculation of the polynomial of (the plat closure of) a link $L$ on
$2N$ strands at any fixed root of unity is shown to be bounded from
above by a linear function of the number of crossings of the link, on
the one hand, and polynomially bounded in terms of the braid index $2N$,
on the other. The growth rate of the time complexity function in terms
of the integer $k$ appearing in the root of unity $q$ can be estimated
to be (polynomially) bounded by resorting to the field theoretical
background given by the ChernSimons theory.
On the quantum hardness of solving
isomorphism problems as nonabelian hidden shift problems (pp504521)
A.M. Childs and P. Wocjan
We consider an approach to deciding isomorphism of rigid
nvertex graphs (and related isomorphism problems) by solving a
nonabelian hidden shift problem on a quantum computer using the standard
method. Such an approach is arguably more natural than viewing the
problem as a hidden subgroup problem. We prove that the hidden shift
approach to rigid graph isomorphism is hard in two senses. First, we
prove that \Omega(n) copies of the hidden shift states are necessary to
solve the problem (whereas O(n\log n) copies are sufficient). Second, we
prove that if one is restricted to singleregister measurements, an
exponential number of hidden shift states are required.
Probability estimates for Shor's
algorithm (pp522550)
P.S. Bourdon and H.T. Williams
Let N be a (large) positive integer, let b
be an integer satisfying 1< b < N that is relatively prime
to N, and let r be the order of b modulo N.
Finally, let QC be a quantum computer whose input register has the size
specified in Shor's original description of his orderfinding algorithm.
In this paper, we analyze the probability that a single run of the
quantum component of the algorithm yields useful informationa
nontrivial divisor of the order sought. We prove that when Shor's
algorithm is implemented on QC, then the probability P of
obtaining a (nontrivial) divisor of $r$ exceeds $.7$ whenever N \ge
2^{11} and r\ge 40, and we establish that $.7736$ is an
asymptotic lower bound for $P$. When $N$ is not a power of an odd prime,
Gerjuoy has shown that $P$ exceeds 90 percent for $N$ and $r$
sufficiently large. We give easily checked conditions on N and
r for this 90 percent threshold to hold, and we establish an
asymptotic lower bound for $P$ of 2\Si(4\pi)/\pi ~0.9499 in this
situation. More generally, for any nonnegative integer $q$, we show that
when QC(q) is a quantum computer whose input register has $q$
more qubits than does QC, and Shor's algorithm is run on QC(q),
then an asymptotic lower bound on P is 2\Si(2^{q+2}\pi)/\pi
(if N is not a power of an odd prime). Our arguments are
elementary and our lower bounds on P are carefully justified.
Quantum cloning of identical mixed
qubits (pp551558)
H.
Fan, B.Y. Liu, and K.J. Shi
Quantum cloning of two identical mixed qubits $\rho \otimes
\rho$ is studied. We propose the quantum cloning transformations not
only for the triplet (symmetric) states but also for the singlet (antisymmetric)
state. We can copy these two identical mixed qubits to $M$ ($M\ge 2$)
copies. This quantum cloning machine is optimal in the sense that the
shrinking factor between the input and the output single qubit achieves
the upper bound. The result shows that we can copy two identical mixed
qubits with the same quality as that of two identical pure states.
Efficient quantum algorithms for the
hidden subgroup problem over semidirect product groups (pp559570)
Y.
Inui and F. Le Gall
In this paper, we consider the hidden subgroup problem (HSP)
over the class of semidirect product groups $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_q$,
for $p$ and $q$ prime. We first present a classification of these groups
in five classes. Then, we describe a polynomialtime quantum algorithm
solving the HSP over all the groups of one of these classes: the groups
of the form $\mathbb{Z}_{p^r}\rtimes\mathbb{Z}_p$, where $p$ is an odd
prime. Our algorithm works even in the most general case where the group
is presented as a blackbox group with not necessarily unique encoding.
Finally, we extend this result and present an efficient algorithm
solving the HSP over the groups $\mathbb{Z}^m_{p^r}\rtimes\mathbb{Z}_p$.
Book Review:
On “Protecting information:
from classical error correction to quantum cryptography (authored by S.
Loepp and W. Wootters)” (pp571572)
G.J.
Milburn
back to QIC online Front page
