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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

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
• Theoretical part of quantum computing is developed substantially
• Practical implementation is still a big problem
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
• 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
• 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
• 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 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 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
• 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 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

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
• 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

|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

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

=

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

Maximally entangled states (Bell's basis)

Is the following state entangled?

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
• 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
• 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
• 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…
• 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
• 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)

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
• 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
• 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
• 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
• 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
• 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
• Is the following state entangled?
• What happens if we apply twice
• negation?
• Hadamard gate?
Thank You!
• http://www.nipissingu.ca/numeric
• http://www.nipissingu.ca/faculty/alexandk/popular/popular.html