Vol.6 No.1
January 1, 2006
Research Articles:
Quantum computer
with dipoledipole interacting twolevel systems
(pp001015)
D. Petrosyan and G. Kurizki
A scalable, highperformance quantum processor can be
implemented using nearresonant dipoledipole interacting dopants in a
transparent solid state host. In this scheme, the qubits are represented
by ground and subradiant states of effective dimers formed by pairs of
closely spaced twolevel systems, while the twoqubit entanglement
either relies on the coherent excitation exchange between the dimers or
is mediated by external laser fields.
Quantum measurements and entropic bounds
(pp016045)
A. Barchielli and G. Lupieri
While a positive operator valued measure gives the
probabilities in a quantum measurement, an instrument gives both the
probabilities and the a posteriori states. By interpreting the
instrument as a quantum channel and by using the monotonicity theorem
for relative entropies many bounds on the classical information
extracted in a quantum measurement are obtained in a unified manner. In
particular, it is shown that such bounds can all be stated as
inequalities between mutual entropies. This approach based on channels
gives rise to a unified picture of known and new bounds on the classical
information (the bounds by Holevo, by Shumacher, Westmoreland and
Wootters, by Hall, by Scutaru, a new upper bound and a new lower one).
Some examples clarify the mutual relationships among the various bounds.
Quantum lower bounds for fanout
(pp046057)
M.
Fang, S. Fenner, F. Green, S. Homer, and Y. Zhang
We consider the resource bounded quantum circuit model
with circuits restricted by the number of qubits they act upon and by
their depth. Focusing on natural universal sets of gates which are
familiar from classical circuit theory, several new lower bounds for
constant depth quantum circuits are proved. The main result is that
parity (and hence fanout) requires log depth quantum circuits, when the
circuits are composed of single qubit and arbitrary size Toffoli gates,
and when they use only constantly many ancill\ae. Under this constraint,
this bound is close to optimal. In the case of a nonconstant number $a$
of ancill\ae\ and $n$ input qubits, we give a tradeoff between $a$ and
the required depth, that results in a nonconstant lower bound for
fanout when $a = n^{1o(1)}$. We also show that, regardless of the
number of ancill\ae\, arbitrary arity Toffoli gates cannot be simulated
exactly by a constant depth circuit family with gates of bounded arity.
A
discrete local invariant for quantum gates
(pp058066)
L. Koponen, V. Bergholm, and M.M.
Salomaa
In this paper we study the properties of twoqubit gates.
We review the most common parameterizations for the local equivalence
classes of twoqubit gates and the connections between them. We then
introduce a new discrete local invariant, namely the number of local
degrees of freedom that a gate can bind. The value of this invariant is
calculated analytically for all the local equivalence classes of twoqubit
gates. We find that almost all twoqubit gates can bind the full six
local degrees of freedom and are in this sense more effective than the
controlledNOT gate which only can bind four local degrees of freedom.
A new algorithm for producing quantum
circuits using KAK
decompositions
(pp067080)
M.Y.
Nakajima, Y. Kawano, and H. Sekigawa
We provide a new algorithm that translates a unitary
matrix into a quantum circuit according to the G=KAK theorem in
Lie group theory. With our algorithm, any matrix decomposition
corresponding to typeAIII KAK decompositions can be
derived according to the given Cartan involution. Our algorithm
contains, as its special cases, CosineSine decomposition (CSD) and
KhanejaGlaser decomposition (KGD) in the sense that it derives the same
quantum circuits as the ones obtained by them if we select suitable
Cartan involutions and square root matrices. The selections of Cartan
involutions for computing CSD and KGD will be shown explicitly. As an
example, we show explicitly that our method can automatically reproduce
the wellknown efficient quantum circuit for the $n$qubit quantum
Fourier transform.
Mini
Tutorial:
The SolovayKitaev algorithm
(pp081095)
C.M. Dawson and M.A. Nielsen
This pedagogical review presents the proof of the
SolovayKitaev theorem in the form of an efficient classical algorithm
for compiling an arbitrary singlequbit gate into a sequence of gates
from a fixed and finite set. The algorithm can be used, for example, to
compile Shor's algorithm, which uses rotations of $\pi / 2^k$, into an
efficient faulttolerant form using only Hadamard, controlled{\sc not},
and $\pi / 8$ gates. The algorithm runs in $O(\log^{2.71}(1/\epsilon))$
time, and produces as output a sequence of $O(\log^{3.97}(1/\epsilon))$
quantum gates which is guaranteed to approximate the desired quantum
gate to an accuracy within $\epsilon > 0$. We also explain how the
algorithm can be generalized to apply to multiqubit gates and to gates
from SU(d).
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