Quantum ComputingQuantum Computing
Oscar Montiel
We present concise information of the main topics that are needed to
understand QC. In the introduction the great potential of this fi eld,
in special for soft computing, is presented. We depart from classical
computing explaining the limitations of classical Turing machines
with the aim to introduce the Quantum Turing Machine to simulate
Quantum algorithms; we use digital logic to introduce the necessary
concepts about the circuit model and reversible computing, with the
purpose of extending these concepts to quantum gates and algorithms.
Brief overviews of the mathematics used in QC and the basic principles
of quantum mechanics also are included. Basic concepts such as the
Bloch sphere, quantum bits, registers, gates, circuits and measurement
are explained.
15.1 Introduction
In the twenty-fi rst century, many signifi cant developments in science
and engineering have been achieved through interdisciplinary research.
Quantum computing (QC) is about computing with quantum systems,
called quantum computers. A quantum computer is a computational
device that makes use of the quantum mechanics (QM) phenomena such
as entanglement, superposition and interference to perform operations
on data. Quantum computers are different from classical binary digital
Instituto Politécnico Nacional, CITEDI, Tijuana, México.
E-mail: oross@ipn.mx
Quantum ComputingQuantum Computing 307
computers since they work with quantum bits. The theoretical model is
the quantum turing machine (QTM) which is a normal turing machine
(TM) with quantum parallelism. Richard Feynman introduced this fi eld
for the fi rst time in the 80’s (Feynman 1982) since he observed that certain
quantum-mechanical effects cannot be simulated in an effi cient way using
a classical computer. Such observation let to speculate that using quantum
effects may become more effi cient computing. The interest grew up in 1994
when Peter Shor published the paper entitled “Algorithms for Quantum
Computation : Discrete Logarithms and Factoring” (Shor 1994) where
he described polynomial time quantum algorithms for fi nding discrete
logarithms and factoring integers.
Quantum computing is a computational model that may provide
exponential advantages over classical computational models such as
probabilistic and deterministic turing machines (Arora and Barak 2010). It
combines QM, information theory and aspects of computer sciences; QC
differs from classical computing based on traditional logic in many respects,
for example, instead of operating on bits and Boolean logic, QC is based
on quantum logic and operates on qubits. The idea is to exploit quantum
effects to compute in an effi cient and faster way than using conventional
computers. QC uses a specifi c implementation to gain a computation
advantage over conventional computers. Although the fi eld is relatively
new, it promises a dramatic computing speed increase using properties
such as superposition and entanglement; it is expected to obtain, in some
cases, an exponential amount of parallelism.
There are several recent works focussing on the application of QC in the
soft computing area, for example: In (Visintin et al. 2012) there is a proposal
to associate the states of a quantum register with membership functions
of fuzzy subsets, and the rules in the fuzzyfi cation process, are performed
using unitary quantum transformations; also it is presented a proposal of
t-norms and t-conorms based on unitary and controlled quantum gates.
With respect to the use of QC with neural networks, in (Nayak et al. 2011)
it is presented the basic concepts of quantum neural computing as well as a
mathematical model of a quantum neuron to discuss the training algorithm.
It is discussed that Quantum Neural Network is a new paradigm based on
the combination of classical neural networks and quantum computation,
and it is presented an introductory representation of quantum artifi cial
neural network modeled on the basis of double-slit experiment. In (Finch
2012) the author proposes two algorithms for training neural nets on a
quantum computer , through a simulation it is shown that they can be
used to train neural nets much faster than using the same methods on a
classical computer. In the robotics fi eld we can fi nd other applications of
quantum computing like an improved hierarchical Q-learning algorithm
in conjunction with quantum parallelization computation for a mobile-

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