quantum computer1

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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

quantum computer1

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