Alexandria ACM SC | Quantum Computing Lecture

7/29/2019 Alexandria ACM SC | Quantum Computing Lecture.

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Introduction to Quantum ComputingAn introduction to non-physicists.


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Why is QuantumComputing? (The

problem)


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Polynomial Timevs

Exponential Time


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Get the feeling ofExponential power: ATower to the Moon!

Distance from Earth to the Moon = 384,400 km

Paper thinness = 0.5 mm

How many folds do we need to make a tower to the moon?


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0.5 * 10^-3 * 2^n = 384,400 * 10^3

n = log(7688*10^8) = 39.5 ~ 40


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Hard problems (NP problems)

Traveling salesman problem (TSP)

Integer factorization problem


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Parallel processing!


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Motivation


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Richard Feynmans observation that certain

quantum mechanical effects cannot be simulatedefficiently on a computer led to speculation that

computation in general could be done moreefficiently if it used these quantum effects.


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This speculation proved justified

when Peter Shor described apolynomial time quantum algorithm

for factoring integers.


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Additional interest in the subject has been created bythe invention of quantum key distribution

and, more recently, popular press accounts ofexperimental successes in quantum teleportation andthe demonstration of a three-bit quantum computer.


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Quantum computers

enables exponentialparallelism


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In quantum systems, the computational spaceincreases exponentially with the size of the system

which enables exponential parallelism. Thisparallelism could lead to exponentially fasterquantum algorithms than possible classically.


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This means that quantum computers will be veryeffective at performing tasks -- like vision

recognition, medical diagnosis, and other forms ofartificial intelligence processing -- that can depend

on very complex pattern matching activities waybeyond the capabilities of classical computers.


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Quantum Mechanics


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Quantum mechanics (QM also known asquantum physics, or quantum theory) is a

branch of physics which deals with physicalphenomena at microscopic scales, where the

action is on the order of the Planck constant.Quantum mechanics departs from classicalmechanics primarily at the quantum realm of

atomic and subatomic length scales.


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Quantum computers may use a quantummechanics phenomena like the spin direction of a

single atom to represent the state of a single unit ofquantum information (qubit), or alternatively the

spin direction of a single electron or the polarizationorientation of a photon.


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Due to laws of quantum mechanics, individualqubits can represent a value of "1", "0" or both

numbers simultaneously. This is because the sub-atomic particles used as qubits can exist in more

than one state -- or "superposition" -- at exactly thesame point in time.


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Anyone who is not shocked by

quantum theory has notunderstood it!

-Niels Bohr


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Qubit


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A qubit or quantum bit is a unit of quantum

informationthe quantum analogue of theclassical bit. A qubit is a two-state quantum-

mechanical system, such as the polarization ofa single photon: here the two states are vertical

polarization and horizontal polarization. In aclassical system, a bit would have to be in one

state or the other, but quantum mechanicsallows the qubit to be in a superposition of both

states at the same time, a property which isfundamental to quantum computing.


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Quantumsuperposition


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Quantum superposition is a

fundamental principle of quantummechanics that holds that a physicalsystemsuch as an electronexists

partly in all its particular theoreticallypossible states (or, configuration ofits properties) simultaneously; but

when measured or observed, it givesa result corresponding to only one ofthe possible configurations.


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There is no good classical explanation of

superpositions: a quantum bit representing 0 and1 can neither be viewed as between 0 and 1 norcan it be viewed as a hidden unknown state thatrepresents either 0 or 1 with a certain probability.


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Multiple Qubits


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A register of n qubits can be in a superposition of

all 2^n possible values. The extra states thathave no classical analog and lead to the

exponential size of the quantum state space are theentangled states.


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The input to a quantum computation can be put in asuperposition state that encodes all possible input

values. Performing the computation on this initialstate will result in superposition of all of the

corresponding output values. This process isknown as quantum parallelism.


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Measurement


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While a quantum system can perform massive

parallel computation, access to the results ofthe computation is restricted. Accessing theresults is equivalent to making a measurement,

which disturbs the quantum state.


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Measuring the output states will randomlyyield only one of the values in the

superposition, and at the same time destroyall of the other results of the computation.


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The process of directly observing a qubit will

actually cause its state to "collapse" to oneor other of its superpositions. In practice thismeans that, when data is read from a qubit,

the result will be either a "1" or a "0".


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This can be imagined as if we have a hugenumber of parallel threads, but we can onlyread the result of one parallel thread, andbecause measurement is probabilistic, we

cannot even choose which one we get.


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Challenges inManufacturing Quantum

Computers


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The greatest problem for building quantum

computers is decoherence, the distortion ofthe quantum state due to interaction withthe environment. For some time it was

feared that quantum computers could not

be built because it would be impossible toisolate them sufficiently from the external

environment. The breakthrough came from

the algorithmic rather than the physicalside, through the invention of quantumerror correction techniques.


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The second drawback, which

is related to the first, is thatmanipulating atoms is difficult!Trying to force one atom to

spin a certain way requirespowerful yet precise magnetic

forces to be applied to theatom so that it doesnt disturbthe other atoms around it.


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Finally, the last draw back is

that you cannot directlymeasure the spin of the atomdue to Heisenbergs

uncertainty principle. As soonas a measurement is taken of

the atom to determine its spin,the spin will change and maynot be the correct answer!


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Realizationtechniques


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Ion Trap: In an ion trap quantum computer:a linear sequence of ions representing the

qubits are confined by electric fields.Lasers are directed at individual ions to

perform single bit quantum gates.

NMR: The nuclear magnetic resonance(NMR) approach has the advantage that it

will work at room temperature, and that

NMR technology in general is already fairlyadvanced.

Superconductors: Needs supercooling


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Realization ofquantum computers


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1995 - Christopher Monroe and David Wineland at NIST (Boulder,Colorado) experimentally realize the first quantum logic gate the C-NOTgate with trapped ions, according to Cirac and Zoller's proposal.

1998 - A working 2-qubit NMR quantum computer used to solve Deutsch'sproblem was demonstrated by Jonathan A. Jones and Michele Mosca atOxford University and shortly after by Isaac L. Chuang at IBM's AlmadenResearch Center together with coworkers at Stanford University and MIT.

1998 - First working 3-qubit NMR computer.

2000 - First working 5-qubit NMR computer demonstrated at the TechnicalUniversity of Munich.

2000 - First execution of order finding (part of Shor's algorithm) at IBM'sAlmaden Research Center and Stanford University.

2000 - First working 7-qubit NMR computer demonstrated at the LosAlamos National Laboratory.


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2006 - First 12-qubit quantum computer

benchmarked.

2007 - a Canadian company called D-Waveannounced what it described as "the world'sfirst commercially viable quantum computer".This was based on a 16-qubit processor -- theRainer R4.7 -- made from the rare metal niobiumsupercooled into a superconducting state.

2009 - Google collaborates with D-WaveSystems on image search technology usingquantum computing.


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2011 - D-Wave launched a fully-commercial, 128-qubit quantumcomputer. Called the D-Wave One, this is described by the company as a

"high performance computing system designed for industrial problemsencountered by fortune 500 companies, government and academia". TheD-Wave One's super-cooled 128-qubit processor is housed inside acryogenics system within a 10 square meter shielded room.

At launch, the D-Wave One cost $10 million. The first D-Wave One was

sold to US aerospace, security and military giant Lockheed Martin in May2011.

2013 - D-Wave launched D-Wave Two system which is a superconducting512-qubit processor chip housed inside a cryogenics system within a 10square meter shielded room.

The computational basis of 500 qubits, for example, would already be toolarge to be represented on a classical computer because it would require2^500 complex values (2^501 bits) to be stored. (For comparison, aterabyte of digital information is only 2^40 bits.)


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Mathematical Model


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Qubit

A quantum bit, or qubit, is a unit vector in a twodimensional complex vector space for

which a particular basis, denoted by {|0>, |1>}, hasbeen fixed. The orthonormal basis |0> and |1> maycorrespond to the vertical and horizontal

polarizations of a photon respectively, or to the

spin-up and spin-down states of an electron.


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Superposition

For the purposes of quantum computation, the basisstates |0> and |1> are taken to represent

the classical bit values 0 and 1 respectively. Unlikeclassical bits however, qubits canbe in a superposition of |0> and |1> such as a|0>+b|

1> where a and b are complex numbers

such that |a|^2 + |b|^2 = 1.


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Measurement

The measurement postulate of quantum mechanicsstates that any device measuring a 2-

dimensional system has an associated orthonormal

basis with respect to which the quantummeasurement takes place. Measurement of a statetransforms the state into one of the

measuring devices associated basis vectors. The

probability that the state is measured asbasis vector |u> is the square of the norm of theamplitude of the component of the original

state in the direction of the basis vector |u>.


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MeasurementIf a superposition a|0>+b|1> is measured with respect to

the basis {|0>, |1>}, the probability that the measuredvalue is |0> is |a|^2 and the probability that the

measured value is |1> is |b|^2.

As measurement changes the state, one cannotmeasure the state of a qubit in two different bases.

Furthermore, quantum states cannot be cloned so it isnot possible to measure a qubit in two ways, evenindirectly by, say, copying the qubit and measuring the

copy in a different basis from the

original.


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Measurement

is equivalentto projection

onto the basis


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Dirac notation

The orthonormal basis {|0>, |1>} can be expressed as{(1, 0)T , (0, 1)T}.

|0>, |1> are called kets (one is ket)

Any complex linear combination of |0> and |1>, a|0> +b|1>, can be written (a, b)T .

bra|ket Dirac notation


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Dirac notation

Combining as in , also writtenas , denotes the inner product of

the two vectors.

For instance, since |0> is a unit vector we have = 1 and since |0> and |1> are orthogonal

we have = 0.

The overlap expression is typically

interpreted as the probability amplitude for the

state to collapse into the state.

Dirac notation


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Dirac notation

The notation |x> and

to |0> and |0> to (0, 0)T.


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Quantum Key Distribution


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Entanglement


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Quantum entanglement is a physical

phenomenon that occurs whenparticles such as photons, electrons,

or molecules interact and then become

separated. Before the interaction eachparticle is described by its ownquantum state. After the interaction the

pair can still be described with adefinite quantum state but eachmember of the pair must also bedescribed relative to one another.


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In classical physics, the possiblestates of a system of n particles,whose individual states can bedescribed by a vector in a two

dimensional vector space, form avector space of 2n dimensions.However, in a quantum system

the resulting state space is muchlarger; a system of n qubits has astate space of 2^n dimensions.


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The state space for two

qubits, each with basis{|0>, |1>}, has basis

{|0>|0>, |0>|1>, |1>|0>,

|1>|1>}

which can be written morecompactly as

{|00>, |01>, |10>, |11>}.


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The state |00>+|11> is an

example of a quantum state thatcannot be described in terms

of the state of each of its

components (qubits) separately.In other words, we cannot find

a1, a2, b1, b2 such that(a1|0> + b1|1>)

(a2|0> + b2|1>) = |00> + |11>.


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If the quantum state of a

pair of particles is in adefinite superposition, and

that superposition cannotbe factored out into the

product of two states (onefor each particle), thenthat pair is entangled.


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When a measurement is madeon one member of such a pair

and the outcome is known (e.g.,clockwise spin), the other

member of this entangled pair isat any subsequent time alwaysfound (when measured) to have

taken the appropriatelycorrelated value (e.g.,

counterclockwise spin).


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There is a correlation between the results ofmeasurements performed on entangled pairs, and thiscorrelation is observed even though the entangled pair

may be separated by arbitrarily large distances.

In the formalism of quantum theory, this effect ofmeasurement happens instantaneously.

Repeated experiments have verified that this workseven when the measurements are performed morequickly than light could travel between the sites of

measurement.

Research into quantum entanglement was initiated bya 1935 paper by Albert Einstein, Boris Podolsky, and

Nathan Rosen describing the EPR paradox.

EPR Paradox


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EPR Paradox


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Quantum Gates


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a quantum gate (or quantumlogic gate) is a basic

quantum circuit operating

on a small number of qubits.They are the building blocks

of quantum circuits, likeclassical logic gates are forconventional digital circuits.

U lik l i l l i


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Unlike many classical logic gates,quantum logic gates are reversible.

However, classical computing canbe performed using only reversiblegates. For example, the reversible

Toffoli gate can implement allBoolean functions. This gate has adirect quantum equivalent, showing

that quantum circuits can performall operations performed by

classical circuits.

Q


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Quantum logic gates arerepresented by unitary matrices.

The most common quantumgates operate on spaces of one

or two qubits, just like thecommon classical logic gatesoperate on one or two bits. This

means that as matrices, quantumgates can be described by 2 2

or 4 4 unitary matrices.


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Toffoli Gate


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Swap Gate


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CNOT Gate

Hadamard Gate (Generates a


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Hadamard Gate (Generates asuperposition of all 2^n possible

states when applied on an n qubit)

Dense Coding


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Dense Coding


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Teleportation


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Teleportation


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Quantum Algorithms


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Quantum Algorithms


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Quantum Algorithms

Quantum algorithms is formed of state encoding, a sequence oftransformations and then a measurement.

all classical algorithms can also be performed on a quantum

computer.

All problems which can be solved on a quantum computer canbe solved on a classical computer. What makes quantumalgorithms interesting is that they might be able to solve someproblems faster than classical algorithms.

The most well known algorithms are Shor's algorithm for factoring,and Grover's algorithm for searching an unstructured database oran unordered list.


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Shor's AlgorithmOn a quantum computer, to factor an integer N, Shor's

algorithm runs in polynomial time (the time taken ispolynomial in log N, which is the size of the input).

Specifically it takes time O((log N)^3), demonstrating thatthe integer factorization problem can be efficiently solved

on a quantum computer and is thus in the complexity classBQP. This is substantially faster than the most efficient

known classical factoring algorithm, the general numberfield sieve, which works in sub-exponential time about

O(e^[1.9 (log N)1/3 (log log N)2/3]).


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What is the relationship between BQP and NP?

BQP (bounded error quantum polynomial time)


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Quantum Algorithms

Grover's algorithm is a quantum algorithm for searching an

unsorted database with N entries in O(N^1/2) time andusing O(log N) storage space.

Post-Quantum Cryptography algorithms! (Lattice-based

cryptography, McEliece cryptosystem)

Conclusions


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Conclusions

Quantum mechanics phenomena can be used in performing computations.

Quantum computers enables exponential parallelism.

Quit can be 0, 1, or in a superposition of both.

Multiple qubits can be in entangled state encoding all possible values.

Measuring a quit in a superposition state return only one possible state and destroy all other possibilities.

Quantum computation has many applications in both communications And computations.

The elements of the theory of quantum computation are already established. Quantum computers, gates,algorithms are ready to be implemented.

Although the realization of real quantum computers faces some technical difficulties, there is a huge

amount of investment in this field from market-leading companies, universities and governments. Thereare already existing commercial implementations.

Within a few number of years, quantum computers may change the way we do computations. It isimportant to prepare yourself for this theory and techniques to be ready for the next wave of computersinshaa Allah.


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Thank you

Documents

Alexandria ACM SC | Quantum Computing Lecture