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