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QUANTUM COMPUTER
BY
JOYCE M THOMAS
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INTRODUCTION
A quantum computer is a device for computationthat makes direct use of quantum mechanicalphenomena, such as superposition and
entanglement, to perform operations on data Quantum computers are different from
traditional computers based on transistors. Thebasic principle behind quantum computation is
that quantum properties can be used torepresent data and perform operations on thesedata
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Large-scale quantum computers could be able
to solve certain problems much faster than
any classical computer by using the best
currently known algorithms, like integer
factorization using Shor's algorithm or thesimulation of quantum many-body systems.
There exist quantum algorithms, such as
Simon's algorithm, which run faster than anypossible probabilistic classical algorithm
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On May 25th, 2011 it was announced that
Lockheed Martin Corporation has entered into
an agreement to purchase the world's first
commercial quantum computing system from
D-Wave Systems Inc.
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Photograph of a chip constructed by D-Wave
Systems Inc., designed to operate as a 128-qubit
superconducting adiabatic quantum optimization
processor, mounted in a sample holder.
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BASIS
A classical computer has a memory made up of bits,where each bit represents either a one or a zero.
While a quantum computer maintains a sequence ofqubits. A single qubit can represent a one, a zero, or,
crucially, any quantum superposition of these. moreover, a pair of qubits can be in any quantum
superposition of 4 states, and 3 qubits in anysuperposition of 8. In general a quantum computerwith n qubits can be in an arbitrary superposition of upto 2n different states simultaneously (this compares toa normal computer that can only be in one of these 2n
states at any one time)
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A quantum computer operates by
manipulating those qubits with a fixed
sequence of quantum logic gates. The
sequence of gates to be applied is called a
quantum algorithm.
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BITS VS QUBITS
Consider first a classical computer that operates on athree-bit register. The state of the computer at anytime is a probability distribution over the 23 = 8different three-bit strings 000, 001, 010, 011, 100, 101,110, 111.
If it is a deterministic computer, then it is in exactlyone of these states with probability 1
If it is a probabilistic computer, then there is apossibility of it being in any one of a number ofdifferent states. We can describe this probabilistic state
by eight nonnegative numbers a,b,c,d,e,f,g,h (where a= probability computer is in state 000, b = probabilitycomputer is in state 001, etc.). There is a restrictionthat these probabilities sum to 1.
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The state of a three-qubit quantum computer is
similarly described by an eight-dimensionalvector (a,b,c,d,e,f,g,h), called a ket.
However, instead of adding to one, the sum ofthe squares of the coefficient magnitudes, | a | 2
+ | b | 2 + ... + | h | 2, must equal one.
Moreover, the coefficients are complex numbers.Since states are represented by complex
wavefunctions, two states being added togetherwill undergo interference, which is a keydifference between quantum computing andprobabilistic classical computing.
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The Bloch sphere is a representation of a qubit
the fundamental building block of quantum
computers
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Qubits are made up of controlled particles and
the means of control (e.g. devices that trap
particles and switch them from one state to
another).
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OPERATION
While a classical three-bit state and a quantum three-qubit stateare both eight-dimensional vectors. They are manipulated quitedifferently for classical or quantum computation.
For computing in either case, the system must be initialized, forexample into the all-zeros string, corresponding to the vector(1,0,0,0,0,0,0,0). In classical randomized computation, the system
evolves according to the application of stochastic matrices, whichpreserve that the probabilities add up to one.
In quantum computation, on the other hand, allowed operationsare unitary matrices, which are effectively rotations . (Exactly whatunitaries can be applied depend on the physics of the quantumdevice.) Consequently, since rotations can be undone by rotating
backward, quantum computations are reversible.
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Finally, upon termination of the algorithm, the result
needs to be read off. In the case of a classical computer, we sample from the
probability distribution on the three-bit register toobtain one definite three-bit string, say 000.
Quantum mechanically, we measure the three-qubit
state, which is equivalent to collapsing the quantumstate down to a classical distribution followed bysampling from that distribution. Note that this destroysthe original quantum state. Many algorithms will onlygive the correct answer with a certain probability,
however by repeatedly initializing, running andmeasuring the quantum computer, the probability ofgetting the correct answer can be increased.
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POTENTIAL I
nteger factorization is believed to be computationallyunfeasible with an ordinary computer for large integers ifthey are the product of few prime numbers (e.g., productsof two 300-digit primes).
By comparison, a quantum computer could efficiently solvethis problem using Shor's algorithm to find its factors.
This ability would allow a quantum computer to decryptmany of the cryptographic systems in use today
In particular, most of the popular public key ciphers arebased on the difficulty of factoring integers. These are usedto protect secure Web pages, encrypted email, and manyother types of data. Breaking these would have significantramifications for electronic privacy and security.
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Consider a problem that has these four properties:
(a)The only way to solve it is to guess answers repeatedly and check them,
(b)The number of possible answers to check is the same as the number of
inputs, (c)Every possible answer takes the same amount of time to check
(d)There are no clues about which answers might be better: generatingpossibilities randomly is just as good as checking them in some specialorder.
An example of this is a password cracker that attempts to guess thepassword for an encrypted file (assuming that the password has amaximum possible length).
For problems with all four properties, the time for a quantum computer tosolve this will be proportional to the square root of the number of inputs.
That can be a very large speedup, reducing some problems from years toseconds. It can be used to attack symmetric ciphers such as Triple DES andAES by attempting to guess the secret key.
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Since chemistry and nanotechnology rely onunderstanding quantum systems, and such
systems are impossible to simulate in an
efficient manner classically, many believe
quantum simulation will be one of the mostimportant applications of quantum
computing.
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REQUIREMENTS FOR A PRACTICAL
QUANTUM COMPUTER
David DiVincenzo, ofIBM, listed the following
requirements for a practical quantum computer:
scalable physically to increase the number of
qubits
qubits can be initialized to arbitrary values
quantum gates faster than decoherence time
universal gate set
qubits can be read easily
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QUANTUM DECOHERENCE
One of the greatest challenges is controlling or
removing quantum decoherence. This usually
means isolating the system from its
environment as the slightest interaction with
the external world would cause the system to
decohere. This effect is irreversible, as it is
non-unitary, and is usually something thatshould be highly controlled
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DEVELOPMENTS
There are a number of quantum computing models, distinguishedby the basic elements in which the computation is decomposed.The four main models of practical importance are
the quantum gate array (computation decomposed into sequenceof few-qubit quantum gates),
the one-way quantum computer (computation decomposed intosequence of one-qubit measurements applied to a highly entangledinitial state (cluster state)),
the adiabatic quantum computer (computation decomposed into aslow continuous transformation of an initial Hamiltonian into a finalHamiltonian, whose ground states contains the solution),
and the topological quantum computer(computation decomposedinto the braiding of anyons in a 2D lattice)
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For physically implementing a quantum computer, many differentcandidates are being pursued, among them (distinguished by thephysical system used to realize the qubits):
Superconductor-based quantum computers (including SQUID-basedquantum computers) (qubit implemented by the state of smallsuperconducting circuits (Josephson junctions))
Trapped ion quantum computer (qubit implemented by the internalstate of trapped ions)
Optical lattices (qubit implemented by internal states of neutralatoms trapped in an optical lattice)
Electrically-defined or self-assembled quantum dots (e.g. the Loss-DiVincenzo quantum computer) (qubit given by the spin states of anelectron trapped in the quantum dot)
Quantum dot charge based semiconductor quantum computer
(qubit is the position of an electron inside a double quantum dot) [ Nuclear magnetic resonance on molecules in solution (liquid-state
NMR) (qubit provided by nuclear spins within the dissolvedmolecule)
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CONCLUSION There exist quantum algorithms, such as Simon's
algorithm, which run faster than any possibleprobabilistic classical algorithm. Given enoughresources, a classical computer can simulate anarbitrary quantum computer. Hence, ignoring
computational and space constraints, a quantumcomputer is not capable of solving any problem whicha classical computer cannot.
However it can be used for specific applications likecipher decoding and the simulation of quantumphysical processes from chemistry and solid statephysics, the approximation of Jones polynomials, andsolving Pell's equation
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