Quantum computing

Alex Karassev

Quantum ComputerQuantum Computer

Quantum computer uses properties of elementary particle that are predicted by quantum mechanics

Usual computers: information is stored in bits

Quantum Computers: information is stored in qubits

Theoretical part of quantum computing is developed substantially

Practical implementation is still a big problem

Quantum computer uses properties of elementary particle that are predicted by quantum mechanics

Usual computers: information is stored in bits

Quantum Computers: information is stored in qubits

Theoretical part of quantum computing is developed substantially

Practical implementation is still a big problem

What is a quantum computer good for?What is a quantum computer good for?

Many practical problems require too much time if we attempt to solve them on usual computers

It takes more then the age of the Universe to factor a 1000-digits number into primes!

The increase of processor speed slowed down because of limitations of existing technologies

Theoretically, quantum computers can provide "truly" parallel computations and operate with huge data sets

Many practical problems require too much time if we attempt to solve them on usual computers

It takes more then the age of the Universe to factor a 1000-digits number into primes!

The increase of processor speed slowed down because of limitations of existing technologies

Theoretically, quantum computers can provide "truly" parallel computations and operate with huge data sets

Probability questionsProbability questions

How many times (in average) do we need to toss a coin to get a tail?

How many times (in average) do we need to roll a die to get a six?

Loaded die: alter a die so that the probability of getting 6 is 1/2.

How many times (in average) do we need to toss a coin to get a tail?

How many times (in average) do we need to roll a die to get a six?

Loaded die: alter a die so that the probability of getting 6 is 1/2.

Quantum computers and probabilityQuantum computers and probability

When the quantum computer gives you the result of computation, this result is correct only with certain probability

Quantum algorithms are designed to "shift" the probability towards correct result

Running the same algorithm sufficiently many times you get the correct result with high probability, assuming that we can verify whether the result is correct or not

The number of repetition is much smaller then for usual computers

When the quantum computer gives you the result of computation, this result is correct only with certain probability

Quantum algorithms are designed to "shift" the probability towards correct result

Running the same algorithm sufficiently many times you get the correct result with high probability, assuming that we can verify whether the result is correct or not

The number of repetition is much smaller then for usual computers

Short HistoryShort History

1970-е: the beginning of quantum information theory

1980: Yuri Manin set forward the idea of quantum computations

1981: Richard Feynman proposed to use quantum computing to model quantum systems. He also describe theoretical model of quantum computer

1985: David Deutsch described first universal quantum computer

1994: Peter Shor developed the first algorithm for quantum computer (factorization into primes)

1970-е: the beginning of quantum information theory

1980: Yuri Manin set forward the idea of quantum computations

1981: Richard Feynman proposed to use quantum computing to model quantum systems. He also describe theoretical model of quantum computer

1985: David Deutsch described first universal quantum computer

1994: Peter Shor developed the first algorithm for quantum computer (factorization into primes)

Short HistoryShort History

1996: Lov Grover developed an algorithm for search in unsorted database

1998: the first quantum computers on two qubits, based on NMR (Oxford; IBM, MIT, Stanford)

2000: quantum computer on 7 qubits, based on NMR (Los-Alamos)

2001: 15 = 3 x 5 on 7- qubit quantum comp. by IBM

2005-2006: experiments with photons; quantum dots; fullerenes and nanotubes as "particle traps"

2007: D-Wave announced the creation of a quantum computer on 16 qubits

1996: Lov Grover developed an algorithm for search in unsorted database

1998: the first quantum computers on two qubits, based on NMR (Oxford; IBM, MIT, Stanford)

2000: quantum computer on 7 qubits, based on NMR (Los-Alamos)

2001: 15 = 3 x 5 on 7- qubit quantum comp. by IBM

2005-2006: experiments with photons; quantum dots; fullerenes and nanotubes as "particle traps"

2007: D-Wave announced the creation of a quantum computer on 16 qubits

Quantum systemQuantum system

Quantum system is a system of elementary particles (photons, electrons, or nucleus) governed by the laws of quantum mechanics

Parameters of the system may include positions of particles, momentum, energy, spin, polarization

The quantum system can be characterized by its state that is responsible for the parameters

The state can change under external influence

fields, laser impulses etc.

measurements

Quantum system is a system of elementary particles (photons, electrons, or nucleus) governed by the laws of quantum mechanics

Parameters of the system may include positions of particles, momentum, energy, spin, polarization

The quantum system can be characterized by its state that is responsible for the parameters

The state can change under external influence

fields, laser impulses etc.

measurements

Some quantum mechanicsSome quantum mechanics

Superposition: if a system can be in either of two states, it also can be in superposition of them

Some parameters of elementary particles are discrete (energy, spin, polarization of photons)

Changes are reversible

The parameters are undetermined before measurements

The original state is destroyed after measurement

No Cloning Theorem: it is impossible to create a copy of unknown state

Quantum entanglement and quantum teleportation

Superposition: if a system can be in either of two states, it also can be in superposition of them

Some parameters of elementary particles are discrete (energy, spin, polarization of photons)

Changes are reversible

The parameters are undetermined before measurements

The original state is destroyed after measurement

No Cloning Theorem: it is impossible to create a copy of unknown state

Quantum entanglement and quantum teleportation

QubitQubit

Qubit is a unit of quantum information

In general, one qubit simultaneously "contains" two classical bits

Qubit can be viewed as a quantum state of one particle (photon or electron)

Qubit can be modeled using polarization, spin, or energy level

Qubit can be measured

As the result of measurement, we get one classical bit: 0 or 1

Qubit is a unit of quantum information

In general, one qubit simultaneously "contains" two classical bits

Qubit can be viewed as a quantum state of one particle (photon or electron)

Qubit can be modeled using polarization, spin, or energy level

Qubit can be measured

As the result of measurement, we get one classical bit: 0 or 1

A model of qubitA model of qubit

|ψ = a⟩ 0 |0⟩ + a1 |1⟩

a0 и a1 are complex numbers such that |a0|2 + |a1 |2 =1

|ψ ⟩ is a superposition of basis states |0 ⟩ и |1⟩

The choice of basis states is not unique

The measurement of ψ results⟩in 0 with probability |a0|2 and in 1 with probability |a1|2

After the measurement the qubit collapses into the basis state that corresponds to the result

a0 и a1 are complex numbers such that |a0|2 + |a1 |2 =1

|ψ ⟩ is a superposition of basis states |0 ⟩ и |1⟩

The choice of basis states is not unique

The measurement of ψ results⟩in 0 with probability |a0|2 and in 1 with probability |a1|2

After the measurement the qubit collapses into the basis state that corresponds to the result

12

30

2

1Example:

0

vector (a0,a1 )or

1

1/4

3/4

Several qubitsSeveral qubits

The system of n qubits "contain" 2n classical bits (basis states)

Thus the potential of a quantum computer grows exponentially

We can measure individual qubits in the multi-qubit system

For example, in a two-qubit system we can measure the state of first or second qubit, or both

The results of measurement are probabilistic

After the measurement the system collapses in the corresponding state

The system of n qubits "contain" 2n classical bits (basis states)

Thus the potential of a quantum computer grows exponentially

We can measure individual qubits in the multi-qubit system

For example, in a two-qubit system we can measure the state of first or second qubit, or both

The results of measurement are probabilistic

After the measurement the system collapses in the corresponding state

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31

Example: two qubitsExample: two qubits

Let's measure the first bit:

01002

12

1 11107

3

72

result

922

312

31 probability

|ψ = a⟩ 0 |00⟩ + a1 |01⟩ + a2 |10⟩ + a3 |11⟩

97

2

332

32

0 1

The coefficients changes so that the ratio is the same

Independent qubitsIndependent qubits

A system of two independent qubits(two non-interacting particles):

A system of two independent qubits(two non-interacting particles):

10 23

21 10 3

532

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23

32

23

35

21

32

21

=

11100100 615

33

65

31

Entangled statesEntangled states

10012

12

1

There is no qubits

a0 |0 + a⟩ 1 |1⟩

b0 |0 + b⟩ 1 |1⟩

s.t. the state

could be represented as

a0b0 |00 + a⟩ 0 b1 |01 + a⟩ 1 b0 |10 + a⟩ 1 b1 |11⟩

measure the first bit

0

1

|01⟩

The value ofsecond bit with100% probability

|10⟩

1

0

10012

12

1

ExamplesExamples

1100

1001

21

21

21

21

11100100 33

32

31

31

Maximally entangled states (Bell's basis)

Is the following state entangled?

Quantum TeleportationQuantum Teleportation

Entangled qubits A and B

1100 21

21 A

qubit with unknown statethat Alice wants to send to Bob

makes А and C entangled

some transformations

measures C

Now Bob knowsthe state of B

makes B into C

Now Bob has qubit C

B

Comm

unication channel (

e.g. p

hone)C

Operations on bitsOperations on bits

NOT: NOT(0) =1, NOT(1)=0

OR: 0 OR 0 = 0, 1 OR 0 = 0 OR 1 = 1 OR 1 = 1

AND: 0 AND 0 = 1 AND 0 = 0 AND 1 = 0, 1 AND 1 = 1

XOR (addition modulo two):0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1

What is NOT ( x OR y)?

What is NOT (x AND y)?

NOT (x OR y) = NOT (x) AND NOT (y)

NOT (x AND y) = NOT (x) OR NOT (y)

NOT: NOT(0) =1, NOT(1)=0

OR: 0 OR 0 = 0, 1 OR 0 = 0 OR 1 = 1 OR 1 = 1

AND: 0 AND 0 = 1 AND 0 = 0 AND 1 = 0, 1 AND 1 = 1

XOR (addition modulo two):0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1

What is NOT ( x OR y)?

What is NOT (x AND y)?

NOT (x OR y) = NOT (x) AND NOT (y)

NOT (x AND y) = NOT (x) OR NOT (y)

Classical and quantum computationClassical and quantum computation

Operations AND and OR are not invertible: even if we know the value of one of two bits and the result of the operation we still cannot restore the value of the other bit

Example: suppose x AND y = 0 and y = 0

what is x?

Because of the laws of quantum mechanics quantum computations must be invertible (since the changes of the quantum system are reversible)

Are there such operations?

Yes! E.g. XOR (addition modulo two)

Operations AND and OR are not invertible: even if we know the value of one of two bits and the result of the operation we still cannot restore the value of the other bit

Example: suppose x AND y = 0 and y = 0

what is x?

Because of the laws of quantum mechanics quantum computations must be invertible (since the changes of the quantum system are reversible)

Are there such operations?

Yes! E.g. XOR (addition modulo two)

Linearity and parallel computationsLinearity and parallel computations

Example: let F be a quantum operation that correspond to a function f(x,y) = (x',y'). Then:

Thus one application of F gives a system that contains the results of f on all inputs!

It is enough to know the results on basis states

Matrix representation

Invertibility

Example: let F be a quantum operation that correspond to a function f(x,y) = (x',y'). Then:

Thus one application of F gives a system that contains the results of f on all inputs!

It is enough to know the results on basis states

Matrix representation

Invertibility

)11()10()01()00(

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3210

3210

fafafafa

aaaaF

Some matrices…Some matrices…

A matrix is a table of numbers, e.g.

We can multiply matrices by vectors:

Moreover, we even can multiply matrices!

A matrix is a table of numbers, e.g.

We can multiply matrices by vectors:

Moreover, we even can multiply matrices!

011

654

321

1

8

4

)1(02)1(11

)1(625 14

)1(122 11

1

2

1

011

654

321

Operations on one qubitOperations on one qubit

Quantum NOT

NOT( a0 |0⟩ + a1 |1 ) = ⟩ a0 |1⟩ + a1 |0⟩

Hadamard gate

H( a0 |0⟩ + a1 |1 ) = ⟩ 1/√2 [ (a0 + a1)|0⟩ + (a0 - a1)|0⟩ ]

Quantum NOT

NOT( a0 |0⟩ + a1 |1 ) = ⟩ a0 |1⟩ + a1 |0⟩

Hadamard gate

H( a0 |0⟩ + a1 |1 ) = ⟩ 1/√2 [ (a0 + a1)|0⟩ + (a0 - a1)|0⟩ ]

1

0

1

0

a

a

01

10

a

aNOT

01 1

100

21

21

21

21

1

0

1

0

a

a

1-1

11

2

1

a

aH

0 1

10

Two qubits: controlled NOT (CNOT)Two qubits: controlled NOT (CNOT)

CNOT( a0|00⟩+a1|01 +⟩ a2|10⟩+a3|11 ) = ⟩ a0|00⟩+a1|01 +⟩ a3|11⟩+a2|10⟩

1011

1110

0101

0000

3

2

1

0

3

2

1

0

a

a

a

a

0100

1000

0010

0001

a

a

a

a

CNOT

CNOT (x,y) = (x, x XOR y)= (x, x⊕y)0⊕0=1⊕1=0, 0⊕1=1⊕0=1

How quantum computer worksHow quantum computer works

The routine

Initialization (e.g. all qubits are in state |0⟩

Quantum computations

Reading of the result (measurement)

"Ideal" quantum computer:

must be universal (capable of performing arbitrary quantum operations with given precision)

must be scalable

must be able to exchange data

The routine

Initialization (e.g. all qubits are in state |0⟩

Quantum computations

Reading of the result (measurement)

"Ideal" quantum computer:

must be universal (capable of performing arbitrary quantum operations with given precision)

must be scalable

must be able to exchange data

Quantum algorithmsQuantum algorithms

Shor's algorithm

Factorization into primes

Work in polynomial time with respect to the number of digits in the representation of an integer

Can be used to break RSA encryption

Grover's algorithm

Database search

"Brute force": about N operations where N is the number of records in the database

Grover's algorithm: about operations

Shor's algorithm

Factorization into primes

Work in polynomial time with respect to the number of digits in the representation of an integer

Can be used to break RSA encryption

Grover's algorithm

Database search

"Brute force": about N operations where N is the number of records in the database

Grover's algorithm: about operationsN

ProblemsProblems

Decoherence

Quantum system is extremely sensitive to external environment, so it should be safely isolated

It is hard to achieve the decoherence time that is more than the algorithm running time

Error correction (requires more qubits!)

Physical implementation of computations

New quantum algorithms to solve more problems

Entangled states for data transfer

Decoherence

Quantum system is extremely sensitive to external environment, so it should be safely isolated

It is hard to achieve the decoherence time that is more than the algorithm running time

Error correction (requires more qubits!)

Physical implementation of computations

New quantum algorithms to solve more problems

Entangled states for data transfer

Practical ImplementationsPractical Implementations

The use of nucleus spins and NMR

Electrons spins and quantum dots

Energy level of ions and ion traps

Use of superconductivity

Adiabatic quantum computers

The use of nucleus spins and NMR

Electrons spins and quantum dots

Energy level of ions and ion traps

Use of superconductivity

Adiabatic quantum computers

D-Wave: quantum computer OrionD-Wave: quantum computer Orion

January 19, 2007: D-Wave Systems (Burnaby, British Columbia) announced a creation of a prototype of commercial quantum computer, called Orion

According to D-Wave, adiabatic quantum computer Orion uses 16 qubits and can solve quite complex practical problems (e.g. search a database and solve Sudoku puzzle)

Unfortunately, D-Wave did not disclose any technical details of their computer

This caused a significant criticism among specialists

Recently, the company received 17 millions investments

January 19, 2007: D-Wave Systems (Burnaby, British Columbia) announced a creation of a prototype of commercial quantum computer, called Orion

According to D-Wave, adiabatic quantum computer Orion uses 16 qubits and can solve quite complex practical problems (e.g. search a database and solve Sudoku puzzle)

Unfortunately, D-Wave did not disclose any technical details of their computer

This caused a significant criticism among specialists

Recently, the company received 17 millions investments

HomeworkHomework

Is the following state entangled?

What happens if we apply twice

negation?

Hadamard gate?

Is the following state entangled?

What happens if we apply twice

negation?

Hadamard gate?

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21

21

21

Thank You!Thank You!

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