QUANTUM COMPUTATION.pptx

7/29/2019 QUANTUM COMPUTATION.pptx

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QUANTUM COMPUTATION

BY :

Ankit TripathiMonika Bisla

Sheetal Taneja


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What is Quantum computer ?

It is a computational device that makes use ofquantum mechanical phenomena, such assuperposition and entanglement, to performoperations on data.


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Why Quantum Computers ?

Feynman (1982): theremay be quantum systemsthat cannot be simulatedefficiently on a classicalcomputer.

Deutsch (1985):proposed that machinesusing quantum processesmight be able to performcomputations thatclassical computers canonly perform very poorly.

Problems

A

Quantum

Computer

can

solve

P


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Examples

Information security and e-commerce arebased on the use of NP problems that are notin P.

Quantum algorithms (e.g., Shors factoringalgorithm) require us to reassess the securityof such systems.

Quantum search algorithm provides aquadratic speed over best classical algorithm

Classical: N steps , Quantum: N1/2 steps


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Basics of QuantumMechanics


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Representation of Data :

Qubits A bit of data is represented by a single

electron that is in one of two states denotedby |0> and |1>, or a superposition of the two.

A single bit of this form is known as a qubit.

Physical Interpretation :

State |0>State |1>

Excited

State

Ground

State

Light pulse of

frequency fortime interval t

Electro

n


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Axioms of Quantum

Computation

Superposition Principle : This axiom tells uswhat are the possible states of a givenquantum system.

Measurement Principle : This axiom governshow much information about the state we canaccess.

Unitary Evolution : This axiom governs howthe state of a quantum system evolves with

time.


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Superposition Principle

Suppose we have a k-level quantum system

- k distinguishable or classical states for thesystem.

- Possible classical states:|0>,|1>,|k-1>

Principle : If a system can be in one ofk

states, then it can also be in any linearsuperposition of these k states.


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Linear superposition can

be given as :|> = a0|0>+a1|1>+a2

|2>+ +ak-1|1>+ak|k>

Also,

|a0|2+|a1|

2+|a2|2+ +

|ak|2 = 1

Here, |a0|2

denotes theprobability of systembeing in state i.


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Measurement Axiom

Suppose that our systemis in a state :

|> = a0|0>+a1|1>+

+ak-1|k-1>+ak|k>

Measure: outcome is oneof the k classical states.

Observe |j> withprobability |aj|2

New state: |j>


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Hilbert Space

The quantum state ofthe k-state system canbe written as a k-dimensional vector

which is in thenormalized form.

The normalization onthe complex amplitudesmakes the k-dimensional vectorspace a Hilbert Space.


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Unitary Evolution

Geometrically, a unitary transformation is arigid body rotation of the hilbert space, thusresulting in a transformation that does notchange its length.

A unitary transformation on hilbert space 2

is specified by mapping the basis states|0> and |1> to orthonormal states

|v0> = a|0> + b|1>

|v1> = c|0> + d|1> Unitary transformation preserves angle

between the vectors.


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Linear transformation of hilbert space can be

given by matrix U :

Also, UU* = I

where U* is the conjugate transpose ofthe matrix.


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Quantum Gates

Rotation Gate :This gate rotates the plane by .

Evolution of a Quantum system is necessarilyunitary which be represented as a rotation onHilbert space.


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Bit Flip / NOT gate :

This gate flips a bit from 0 to 1 and vice-

versa.


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Phase Flip :

It is a NOT gate acting in |+>,|-> basis.

Z|+> = |-> and Z|-> = Z|+>


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Hadamard gate :

Can be viewed as a -rotation around /8

axis in the complex plane.


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Also,

H* = H and H2 = I

It converts |0> to |+> and |1> to |->.

i.e. H|0> = |+> and

H|1> = |->


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Relationship between Gates

Circuit :

From this we can deduce that :X = HZH

Z = HXH which can beverified.


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Quantum Circuits

One qubit gate:

0+ 1 0 +

1

2-qubit gate:


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Here, U is a 4* 4 unitary matrix and

U U* = U* U = I


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CNOT Gate

A two-qubit gate. Control bit: First bit

Target bit: Second bit

The target bit flips if and only if the control bitis 1.

Circuit notation:

a a

b a b

a , b { 0,1 }


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Matrix representation:

|00> |00>

|01> |01>|10> |11>

|11> |10>


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Grovers Algorithm

Grover demonstrated that quantumcomputers can perform database searchfaster than their classical counterparts.

Database: Represented by a haystack.

Element to be searched: Needle

Searching an unsorted database takes o(n)time classically but Grovers Algorithm makesit possible in O(n1/2) time.


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Problem Definition

Given

f : {0,1,2,,N-1} {0,1}

Find x: f(x) = 1

Hardest Case: When there

is exactly one x such that

f(x) = 1.


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Steps of Grovers Algorithm

Step 1.Prepare a quantum register to benormalized and uniquely in the first state.

Place the register in an equal

superposition of all states by applying theWalsh Hadamard gate.


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Step 2 : Phase Inversion

Let the system be in any state S.

If f(S) = 0, Leave the state unaltered.else, rotate the phase by radians.


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Step 3 : Inversion about Average

Calculate the average of all the states and

apply inversion about it.


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Implementation of Grovers Algorithm usingQuantum Circuits :

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QUANTUM COMPUTATION.pptx