|
Vol.8
No.1&2 January, 2008
Research Articles:
Feasibility study of free-Space
quantum key distribution in the mid-infrared (pp0001-0011) PDF
G.
Temporao, H. Zibinden, S. Tanzilli, N. Gisin, T. Aellen, M. Giovannini,
J. Faist, and J. von der Weid
We report on a feasibility study of a free-space Quantum
Key Distribution setup operating at a mid-infrared wavelength. Alice
sends polarization-coded pseudo-single photons from a Quantum Cascade
Laser at 4.6 $\mu$m to Bob, who uses a nonlinear crystal and a Silicon
Avalanche Photodiode to perform the detection via Sum Frequency
Generation. Theoretical predictions, based on a proof-of-principle
experiment, show that, under certain foggy atmospheric conditions, the
proposed system is barely affected, whereas other systems operating at
standard near-infrared wavelengths would simply not work at all.
Quantum algorithm design using dynamic
learning (pp0012-0029) PDF
E.C.
Behrman, J.E. Steck, P. Kumar, and K.A. Walsh
We present a dynamic learning paradigm for
``programming'' a general quantum computer. A learning algorithm is used
to find the control parameters for a coupled qubit system, such that the
system at an initial time evolves to a state in which a given
measurement corresponds to the desired operation. This can be thought of
as a quantum neural network. We first apply the method to a system of
two coupled superconducting quantum interference devices (SQUIDs), and
demonstrate learning of both the classical gates XOR and XNOR. Training
of the phase produces a gate similar to the CNOT. Striking out for
somewhat more interesting territory, we attempt learning of an
entanglement witness for a two qubit system. Simulation shows a
reasonably successful mapping of the entanglement at the initial time
onto the correlation function at the final time for both pure and mixed
states. For pure states this mapping requires knowledge of the phase
relation between the two parts; however, given that knowledge, this
method can be used to measure the entanglement of an otherwise unknown
state. The method is easily extended to multiple qubits or to quNits.
$\epsilon$-convertibility of entangled
states and extension of Schmidt rank in infinite-dimensional systems
(pp0030-0052) PDF
M.
Owari, S.L. Braunstein, K. Nemoto, and M. Murao
By introducing the concept of $\epsilon$-convertibility,
we extend Nielsen's and Vidal's theorems to the entanglement
transformation of infinite-dimensional systems. Using an
infinite-dimensional version of Vidal's theorem we derive a new
stochastic-LOCC (SLOCC) monotone which can be considered as an extension
of the Schmidt rank. We show that states with polynomially-damped
Schmidt coefficients belong to a higher rank of entanglement class in
terms of SLOCC convertibility. For the case of Hilbert spaces of
countable, but infinite dimensionality, we show that there are actually
an uncountable number of classes of pure non-interconvertible bipartite
entangled states.
Practical effects in the preparation of
cluster states using weak non-linearities (pp0053-0067) PDF
P.P.
Rohde, W.J. Munro, T.C. Ralph, P. van Loock, and K. Nemoto
We discuss experimental effects in the implementation of
a recent scheme for performing bus mediated entangling operations
between qubits. Here a bus mode, a strong coherent state, successively
undergoes weak Kerr-type non-linear interactions with qubits. A
quadrature measurement on the bus then projects the qubits into an
entangled state. This approach has the benefit that entangling gates are
non-destructive, may be performed non-locally, and there is no need for
efficient single photon detection. In this paper we examine practical
issues affecting its experimental implementation. In particular, we
analyze the effects of post-selection errors, qubit loss, bus loss,
mismatched coupling rates and mode-mismatch. We derive error models for
these effects and relate them to realistic fault-tolerant thresholds,
providing insight into realistic experimental requirements.
Exploring scalar quantum walks on Cayley
graphs (pp0068-0081) PDF
O.L.
Acevedo, J. Roland, and N.J. Cerf
A quantum walk, \emph{i.e.}, the quantum evolution of a
particle on a graph, is termed \emph{scalar} if the internal space of
the moving particle (often called the coin) has dimension one. Here, we
study the existence of scalar quantum walks on Cayley graphs, which are
built from the generators of a group. After deriving a necessary
condition on these generators for the existence of a scalar quantum
walk, we present a general method to express the evolution operator of
the walk, assuming homogeneity of the evolution. We use this necessary
condition and the subsequent constructive method to investigate the
existence of scalar quantum walks on Cayley graphs of groups presented
with two or three generators. In this restricted framework, we classify
all groups -- in terms of relations between their generators -- that
admit scalar quantum walks, and we also derive the form of the most
general evolution operator. Finally, we point out some interesting
special cases, and extend our study to a few examples of Cayley graphs
built with more than three generators.
On the role of shared entanglement
(pp0082-0095) PDF
D.
Gavinsky
Despite the apparent similarity between shared randomness
and shared entanglement in the context of Communication Complexity, our
understanding of the latter is not as good as of the former. In
particular, there is no known ``entanglement analogue'' for the famous
theorem by Newman, saying that the number of shared random bits required
for solving any communication problem can be at most logarithmic in the
input length (i.e., using more than $\asO[]{\log n}$ shared random bits
would not reduce the complexity of an optimal solution). In
this paper we prove that the same is not true for entanglement. We
establish a wide range of tight (up to a polylogarithmic factor)
entanglement vs.\ communication trade-offs for relational problems. The
low end is:\ for any $t>2$, reducing shared entanglement from $log^tn$
to $\aso[]{log^{t-2}n}$ qubits can increase the communication required
for solving a problem almost exponentially, from $\asO[]{log^tn}$ to $\asOm[]{\sqrt
n}$. The high end is:\ for any $\eps>0$, reducing shared entanglement
from $n^{1-\eps}\log n$ to $\aso[]{n^{1-\eps}/\log n}$ can increase the
required communication from $\asO[]{n^{1-\eps}\log n}$ to $\asOm[]{n^{1-\eps/2}/\log
n}$. The upper bounds are demonstrated via protocols which are \e{exact}
and work in the \e{simultaneous message passing model}, while the lower
bounds hold for \e{bounded-error protocols}, even in the more powerful \e{model
of 1-way communication}. Our protocols use shared EPR pairs while the
lower bounds apply to any sort of prior entanglement. We
base the lower bounds on a strong direct product theorem for
communication complexity of a certain class of relational problems. We
believe that the theorem might have applications outside the scope of
this work.
Universal quantum computation with
electronic qubits in decoherence-free subspace
(pp0096-0105) PDF
X.L.
Zhang, M. Feng, and K.L. Gao
We investigate how to carry out universal quantum
computation deterministically with free electrons in decoherence-free
subspace by using polarizing beam splitters, charge detectors, and
single-spin rotations. Quantum information in our case is encoded in
spin degrees of freedom of the electron-pairs which construct a
decoherence-free subspace. We design building blocks for two
noncommutable single-logic-qubit gates and a logic controlled phase
gate, based on which a universal and scalable quantum information
processing robust to dephasing is available in a deterministic way.
Generalized Clifford groups and
simulation of associated quantum circuits (pp0106-0126) PDF
S.
Clark, R. Jozsa, and N. Linden
Quantum computations starting with computational basis states and
involving only Clifford operations, are classically simulable despite
the fact that they generate highly entangled states; this is the content
of the Gottesman-Knill theorem. Here we isolate the ingredients of the
theorem and provide generalisations of some of them with the aim of
identifying new classes of simulable quantum computations. In the usual
construction, Clifford operations arise as projective normalisers of the
first and second tensor powers of the Pauli group. We consider replacing
the Pauli group by an arbitrary finite subgroup $G$ of $U(d)$. In
particular we seek $G$ such that $G\otimes G$ has an entangling
normaliser. Via a generalisation of the Gottesman-Knill theorem the
resulting normalisers lead to classes of quantum circuits that can be
classically efficiently simulated. For the qubit case $d = 2$ we
exhaustively treat all finite irreducible subgroups of $U(2)$ and find
that the only ones (up to unitary equivalence and trivial phase
extensions) with entangling normalisers are the groups generated by X
and the $n^{\rm th}$ root of $Z$ for $n \in \N$.
On the Pauli graphs on N-qudits
(pp0127-0146) PDF
M.
Planat and M. Saniga
A comprehensive graph theoretical and finite geometrical
study of the commutation relations between the generalized Pauli
operators of $N$-qudits is performed in which vertices/points correspond
to the operators and edges/lines join commuting pairs of them. As per
two-qubits, all basic properties and partitionings of the corresponding
{\it Pauli graph} are embodied in the geometry of the generalized
quadrangle of order two. Here, one identifies the operators with the
points of the quadrangle and groups of maximally commuting subsets of
the operators with the lines of the quadrangle. The three
basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin's
array and a bipartite-part and (c) a maximum independent set and the
Petersen graph. These factorizations stem naturally from the existence
of three distinct geometric hyperplanes of the quadrangle, namely a set
of points collinear with a given point, a grid and an ovoid, which
answer to three distinguished subsets of the Pauli graph, namely a set
of six operators commuting with a given one, a Mermin's square, and set
of five mutually non-commuting operators, respectively. The
generalized Pauli graph for multiple qubits is found to follow from
symplectic polar spaces of order two, where maximal totally isotropic
subspaces stand for maximal subsets of mutually commuting operators. The
substructure of the (strongly regular) $N$-qubit Pauli graph is shown to
be pseudo-geometric, i.\,e., isomorphic to a graph of a partial
geometry. Finally, the (not strongly regular) Pauli graph of a two-qutrit
system is introduced; here it turns out more convenient to deal with its
dual in order to see all the parallels with the two-qubit case and its
surmised relation with the generalized quadrangle $Q(4,3)$, the dual of
$W(3)$.
The Jones polynomial: quantum algorithms
and applications in quantum complexity theory (pp0147-0180) PDF
P.
Wocjan and J. Yard
We analyze relationships between quantum computation and
a family of generalizations of the Jones polynomial. Extending recent
work by Aharonov et al., we give efficient quantum circuits for
implementing the unitary Jones-Wenzl representations of the braid group.
We use these to provide new quantum algorithms for approximately
evaluating a family of specializations of the HOMFLYPT two-variable
polynomial of trace closures of braids. We also give algorithms for
approximating the Jones polynomial of a general class of closures of
braids at roots of unity. Next we provide a self-contained proof of a
result of Freedman et al.\ that any quantum computation can be replaced
by an additive approximation of the Jones polynomial, evaluated at
almost any primitive root of unity. Our proof encodes two-qubit
unitaries into the rectangular representation of the eight-strand braid
group. We then give QCMA-complete and PSPACE-complete problems which are
based on braids. We conclude with direct proofs that evaluating the
Jones polynomial of the plat closure at most primitive roots of unity is
a \#P-hard problem, while learning its most significant bit is PP-hard,
circumventing the usual route through the Tutte polynomial and graph
coloring.
back to QIC online Front page
|
