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quantum unitary transformations. We also introduce a simple method for a qubit
measurement and its application to other cases.
Comment: It s not clear which real life problems are concerned by these results.
The paper introduces methods for strange single qubit transformations based ei-
ther on measurement or on unitarity and analytical methods for calculating the
optimal parameters and their fidelity.
1.7.8 The Representation of Numbers in Quantum Mechanics
author(s): P.Benioff
where: quant-ph/0103078
Earlier work on modular arithmetic of k-ary representations of length L of the
natural numbers in quantum mechanics is extended here to k-ary representations
of all natural numbers, and to integers and rational numbers. Since the length L is
indeterminate, representations of states and operators using creation and annihila-
tion operators for bosons and fermions are defined. Emphasis is on definitions and
properties of operators corresponding to the basic operations whose properties are
given by the axioms for each type of number. The importance of the requirement
of efficient implementability for physical models of the axioms is noted. Successor
operations for each value of j corresponding to addition of kj-1 if j >0 and kj if
j
which is the case for all computers, that implementation of the addition and multi-
plication operators, which are defined in terms of polynomially many iterations of
the successors, should be efficient. This conclusion does not follow from definitions
based on the successor for j =1 only.
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1.7.9 Multiparticle entanglement with quantum logic networks: Applica-
tion to cold trapped ions
author(s): M.Sasura, V.Buzek
where: quant-ph/0103067, Phys.Rev.A 64, 012305 (2001)
We show how to construct a multi-qubit control gate on a quantum register of an
arbitrary size N. This gate performs a single-qubit operation on a specific qubit
conditioned by the state of other N - 1 qubits. We provide an algorithm how to
build up an array of networks consisting of single-qubit rotations and multi-qubit
control-NOT gates for the synthesis of an arbitrary entangled quantum state of N
qubits. We illustrate the algorithm on a system of cold trapped ions. This example
illuminates the efficiency of the direct implementation of the multi-qubit CNOT
gate compared to its decomposition into a network of two-qubit CNOT gates.
1.7.10 Efficient Scheme for Initializing a Quantum Register with an Ar-
bitrary Superposed State
author(s): G.L.Long, Y.Sun
where: quant-ph/0104030, accepted by Phys. Rev. A
Preparation of a quantum register is an important step in quantum computation
and quantum information processing. It is straightforward to build a simple quan-
tum state such as |i1i2...in with ij being either 0 or 1, but is a non-trivial task
to construct an arbitrary superposed quantum state. In this Paper, we present a
scheme that can most generally initialize a quantum register with an arbitrary su-
perposition of basis states. Implementation of this scheme requires O(Nn2) stan-
dard 1- and 2-bit gate operations, without introducing additional quantum bits.
Application of the scheme in some special cases is discussed.
Comment: Arbitrary here really means that the N amplitudes are the most gen-
eral complex numbers consistent with the normalization condition. The procedure
requires a classical preprocessing and an execution time which grows exponentially
slow. Sometime the construction can be simplified, but it requires further prepro-
cessing.
2 Quantum algorithms
2.1 Generalities on Algorithms and basic algorithms
2.1.1 Counterfactual Computation
author(s): G.Mitchison, R.Jozsa
where: quant-ph/9907007
Suppose that we are given a quantum computer programmed ready to perform
a computation if it is switched on. Counterfactual computation is a process by
which the result of the computation may be learnt without actually running the
computer. Such processes are possible within quantum physics and to achieve this
effect, a computer embodying the possibility of running the computation must be
available, even though the computation is, in fact, not run. We study the possi-
bilities and limitations of general protocols for the counterfactual computation of
decision problems (where the result r is either 0 or 1). If p(r) denotes the proba-
bility of learning the result r for free in a protocol then one might hope to design
a protocol which simultaneously has large p(0) and p(1). However we prove that
p(0) + p(1) never exceeds 1 in any protocol and we derive further constraints on
p(0) and p(1) in terms of N, the number of times that the computer is not run.
In particular we show that any protocol with p(0) + p(1) = 1 - must have N
tending to infinity as tends to 0. These general results are illustrated with some
explicit protocols for counterfactual computation. We show that interaction-free
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measurements can be regarded as counterfactual computations, and our results
then imply that N must be large if the probability of interaction is to be close to
zero. Finally, we consider some ways in which our formulation of counterfactual
computation can be generalised.
Comment: Speculation about the possibility of obtaining the result of the computa-
tion having always measured the computer to be shut off (i.e. without interaction)
at the cost of spending much more than a single computation run time. It is not
clear which advantage can be gained from such an approach.
2.1.2 Sophisticated quantum search without entanglement
author(s): D.A.Meyer
where: quant-ph/0007070, UCSD preprint 10 (2000)
Although entanglement is widely considered to be necessary for quantum algo-
rithms to improve on classical ones, Lloyd has observed recently that Grover s
quantum search algorithm can be implemented without entanglement, by replac-
ing multiple particles with a single particle having exponentially many states. We
explain that this maneuver removes entanglement from any quantum algorithm.
But all physical resources must be accounted for to quantify algorithm complex-
ity, and this scheme typically incurs exponential costs in some other resource(s).
In particular, we demonstrate that a recent experimental realization requires ex-
ponentially increasing precision. There is, however, a quantum algorithm which
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