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Quantum computing. Alex Karassev. Quantum 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

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quantum computing

Quantum computing

Alex Karassev

quantum computer
Quantum 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
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
probability questions
Probability 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.
quantum computers and probability
Quantum 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
short history
Short 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)
short history7
Short 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 x5 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 system
Quantum system
  • Quantum systemis 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 mechanics
Some 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
qubit
Qubit
  • 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 qubit

vector (a0,a1)

|ψ〉 = a0|0〉 + a1|1〉

A model of qubit
  • a0и a1are 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 ofψ〉 resultsin 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

or

1/4

Example:

3/4

several qubits
Several qubits
  • The system ofn 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
example two qubits

|ψ〉 = a0|00〉 + a1|01〉+a2|10〉 + a3|11〉

Example: two qubits

Let's measure the first bit:

1

0

result

probability

The coefficients changes so that the ratio is the same

independent qubits
Independent qubits

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

=

entangled states
Entangled states

There is no qubitsa0 |0〉 + a1 |1〉b0 |0〉 + b1 |1〉s.t. the state

The value ofsecond bit with100% probability

|01〉

1

0

measure the first bit

1

|10〉

0

could be represented asa0b0 |00〉 + a0 b1 |01〉 + a1 b0 |10〉 + a1 b1 |11〉

examples
Examples

Maximally entangled states (Bell's basis)

Is the following state entangled?

quantum teleportation

A

B

C

Quantum Teleportation

Entangled qubitsA and B

qubit with unknown statethat Alice wants to send to Bob

Now Bob knowsthe state of B

makes А and C entangled

Communication channel (e.g. phone)

makes B into C

some transformations

Now Bob has qubit C

measures C

operations on bits
Operations 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)
classical and quantum computation
Classical and quantum computation
  • OperationsAND andOR 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 andy = 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 computations
Linearity 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
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!
operations on one qubit
Operations on one qubit
  • Quantum NOTNOT( a0 |0〉 + a1 |1〉) = a0 |1〉 + a1 |0〉
  • Hadamard gateH( a0 |0〉 + a1 |1〉) = 1/√2 [ (a0+ a1)|0〉 + (a0- a1)|0〉]
two qubits controlled not cnot
Two qubits: controlled NOT (CNOT)

CNOT (x,y) = (x, x XOR y)= (x, x⊕y)

0⊕0=1⊕1=0, 0⊕1=1⊕0=1

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

how quantum computer works
How 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
quantum algorithms
Quantum 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": aboutN operations where N is the number of records in the database
    • Grover's algorithm: about operations
problems
Problems
  • 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 implementations
Practical 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
d wave quantum computer orion
D-Wave: quantum computer Orion
  • January 19, 2007:D-Wave Systems (Burnaby, British Columbia) announced a creation of a prototype of commercial quantum computer, calledOrion
  • 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
homework
Homework
  • Is the following state entangled?
  • What happens if we apply twice
    • negation?
    • Hadamard gate?
thank you
Thank You!
  • http://www.nipissingu.ca/numeric
  • http://www.nipissingu.ca/faculty/alexandk/popular/popular.html