Quantum computation a review

Journal of Advanced Computing and Communication Technologies (ISSN: 2347 - 2804)
Volume No. 3 Issue No.1, February 2015
23
Quantum Computation: A Review
By
Nidhi Jain1
, Dr. Mahesh Kumar Porwal2
1
M. Tech. Student, 2
Professor
Department of Electronics and Communication Engineering
Shrinathji Institute of Technology & Engineering, Nathdwara (Rajasthan)
nidhijain745@gmail.com, porwal5@yahoo.com
Abstract - In the classical model, the fundamental
building block is represented by bits exists in two states a 0 or
a 1. Computations are done by logic gates on the bits to
produce other bits. By increasing the number of bits, the
complexity of problem and the time of computation increases.
A quantum algorithm is a sequence of operations on a register
to transform it into a state which when measured yields the
desired result. This paper provides introduction to quantum
computation by developing qubit, quantum gate and quantum
circuits.
I. Introduction
In classical computing there are a number of
problems that cannot be solved with efficient
algorithms. For example, the best classical
algorithm for factorization a large integer N
increases exponentially with the size of the
integer. In searching problem, the best classical
algorithm increases directly as the size of the
database. Quantum computation is a new
computational model that utilizes the principle of
quantum physics. It solves the problems more
efficiently in comparison with classical
computation and solves that problems also which
cannot be solved by classical computation.
Qubit is the fundamental unit in quantum
computing. It is similar to a bit, either valued 0 or
1. A qubit can exist in any superposition of the 0
or 1 state simultaneously. When a qubit in such
state is measured the superposition will be
destroyed. A quantum algorithm consists of
sequence operations on a register to transform it
into a state which when measured yields the
desired result with high probability. An n-bit
quantum register can store an exponential amount
of information. The register as a whole can be in
an arbitrary superposition of the 2n
base states
which it can be measured to be in while this
superposition. When a qubit is in this state, it can
be thought of as existing in two universes: as a 0
in one universe and as a 1 in the other. An
operation on such a qubit effectively acts on both
values at the same time. The significant point
being that by performing the single operation on
the qubit, we have performed the operation on
two different values.

24
A quantum computer would consist of many qubit
gates with entangled states. These gates could be
addressed in parallel by unitary transformations,
which must be carried out reversibly, implying no
loss of energy in a gate operation. Quantum
computers are “wired” so that they can do many
calculations at the same time. This is known as
“quantum parallelism” and represents the power of
a quantum computer.
Several systems have been proposed for quantum
computing including photons in nonlinear optical
systems, trapped ions, electron and nuclear spins,
quantum dots, and Josephson junctions. There are
advantages and disadvantages to all these
approaches. Some, such as those employing light
or trapped ions, have demonstrated that they can
provide individual qubits of excellent quality. But,
it is not yet known if they can be scaled up to
produce systems with many qubits and many
possible quantum gate operations. There is nothing
a quantum computer can do that cannot also be
done by an ordinary computer. However, for some
problems, a quantum computer may be many
orders of magnitude faster. There are presently
two important problems involving commerce and
security for which a quantum computer is believed
to be superior. These are finding the factors of a
large number and searching an unstructured
database.
The spectacular promise of quantum computers is
to enable new algorithms which render feasible
problems requiring exorbitant resources for their
solution on a classical computer. At the time of
writing, two broad classes of quantum algorithms
are known which fulfill this promise. The first
class of algorithms is based upon Shor’s quantum
Fourier transform, and includes remarkable
algorithms for solving the factoring and discrete
logarithm problems, providing a striking
exponential speedup over the best known
classical algorithms. The second class of
algorithms is based upon Grover’s algorithm for
performing quantum searching. These provide a
less striking but still remarkable quadratic
speedup over the best possible classical
algorithms. The quantum searching algorithm
derives its importance from the widespread use of
search-based techniques in classical algorithms,
which in many instances allows a straightforward
adaptation of the classical algorithm to give a
faster quantum algorithm.
Fig. 1 The main quantum algorithms and their
relationships, including some notable applications.
Figure 1 sketches the state of knowledge about
quantum algorithms at the time of writing,

25
including some sample applications of those
algorithms. Naturally, at the core of the diagram
are the quantum Fourier transform and the
quantum searching algorithm. Of particular
interest in the figure is the quantum counting
algorithm. This algorithm is a clever combination
of the quantum searching and Fourier transform
algorithms, which can be used to estimate the
number of solutions to a search problem more
quickly than is possible on a classical computer.
II. Quantum Computation model
In the quantum circuit model, we have logical
qubits carried along ‘wires’, and quantum gates
that act on the qubits. A quantum gate acting on n
qubits has the input qubits carried to it by n wires,
and n other wires carry the output qubits away
from the gate. A quantum circuit is often
illustrated schematically by a circuit diagram. The
wires are shown as horizontal lines, and we
imagine the qubits propagating along the wires
from left to right in time. The gates are shown as
rectangular blocks. For example, the 4-qubit state
|ψi_ = |0> ⊗ |0> ⊗ |0> ⊗ |0> enters the circuit at the
left. These qubits are processed by the gates U1,
U2, U3, and U4. At the output of the circuit we
have the collective (possibly entangled) 4-qubit
state |ψf>. A measurement is then made of the
resulting state. The measurement will often be a
simple qubit-by-qubit measurement in the
computational basis, but in some cases may be a
more general measurement of the joint state. A
measurement of a single qubit in the
computational basis is denoted on a circuit
diagram by a small triangle.
Fig. 2 quantum circuit
Fig. 3 A general quantum operation or super
operator can be realized using a unitary operation
by adding an ancillary and tracing out part of the
output.
The triangle symbol will be modified for cases, in
which there is a need to indicate different types of
measurements. The measurement postulate stated
that a measurement outputs a classical label ‘i’
indicating the outcome of the measurement and a
quantum state |φi>. Thus, we could in general
draw our measurement symbol with a ‘quantum’
wire carrying the quantum state resulting from

26
the measurement, together with a classical wire
carrying the classical label.
III. Existing work in Quantum
Computation
Quantum Computation can be used to extract
statistics, such as the minimal element, from an
unordered data set, more quickly than is possible
on a classical computer. It can be used to speed up
algorithms for some problems in NP – specifically,
those problems for which a straightforward search
for a solution is the best algorithm known. Finally,
it can be used to speed up the search for keys to
cryptosystems such as the widely used Data
Encryption Standard (DES). In the early 1980's,
physicist Richard Feynman observed that no
classical computer could simulate quantum
mechanical systems without incurring exponential
slowdown. At the same time, it seems reasonable
that a computer which behaves in a manner
consistent with quantum mechanics could, in
principle, simulate such systems without
exponential slowdown.
For many years the study of quantum computing
was primarily an academic curiosity. Shor (1994)
developed a polynomial time algorithm for
factoring large integers. According to Williams
and Clearwater (1998), it is not known if there is a
classical algorithm for factoring large integers
efficiently, but the best algorithms published thus
far are super-polynomial. This algorithm coupled
with the prominence of cryptographic systems
based on factoring large integers fueled study of
quantum computation, both from an algorithmic
and a manufacturing point of view. Grover
(1996) provided an O(√n) time algorithm for
finding a single marked element in an unsorted
database of n elements. The best possible
classical algorithm would run in O (n) time. This
search problem was not the first problem for
which a quantum computer was shown to be
better than any possible classical computer, but it
was the first problem of real utility found where a
quantum computer outperforms a classical
computer in an asymptotic sense.
While Shor's algorithm may be of more
immediate utility, Grover's algorithm seems more
interesting in a theoretical sense, as it highlights
an area of fundamental superiority in quantum
computation.
IV. References
[1]. Michael A. Nielsen & Isaac L. Chuang, Quantum
Computation and Quantum information, 10th
anniversary edition, Cambridge University press.
[2]. Phillp Kaye, Raymond Laflamme, Michele
Mosca, An Introduction to Quantum Computing,
Oxford University Press
[3]. Quantum computation and grover's algorithm,
aaron krahn

Quantum computation a review

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Quantum computation a review