Quantum computing. Jan Černý, FIT, Czech Technical University in Prague. České vysoké učení technické v Praze. Fakulta informačních technologií

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1 České vysoké učení technické v Praze Fakulta informačních technologií Katedra teoretické informatiky Evropský sociální fond Praha & EU: Investujeme do vaší budoucnosti MI-MVI Methods of Computational Intelligence(2010/2011) Quantum computing Jan Černý, FIT, Czech Technical University in Prague 1

2 Quantum mechanics Describes 3 of the main interactions (Electromagnetic, Strong and Weak nuclear) in the micro-world. Gravity does not play role in the standard cases in micro-world (for example Elecromagneticinteraction is stronger). Border between micro and macro world is not a static boundary, but depends on interactions between given object and surroundings. If there is no interaction (no light, no heat emitted, no particles in the surrounding air) object will behave according to quantum mechanics. Theoretically you can put human into vacuum, cool him to absolute zero and he will start behaving as a wave of light. Largest object so-far showing quantum behavior had tens of carbon atoms. 2

3 Differences of micro-world Discrete values of some variables (energy, momentum..) Wave-particle dualism (objects may behave as particle in some situations and as wave in other situations) Born rule: wave intensity = particle probability Act of measurement affects measured object (ie. state of object changes after measurement) Uncertainty principle (more precisely we measure one dynamic variable, the less precision we get measuring the other one) Complementarity of variables Non-determinism (Two exactly same experiments may end up differently quantum mechanics can tell us probabilities of an outcome, but can t tell us what the outcome will be) 3

4 QM in informatics - Quantum tunneling Situation when particle tunnels through barrier that should normally be impenetrable. With minimizing our electronics we see this behavior more and more often and it stands as a limit to minimize things even further without accounting for quantum mechanics. 4

5 Quantum tunneling Flash memory Erasing a flash-memory cell (resetting to a logical 1) is achieved by applying a voltage across the source and control gate. Electrons are pulled by quantum tunneling through the barrier of the Si0 2 insulator layer surrounding the floating gate. 5

6 Qubit Is unit of quantum information derived from standard bit. It contains not only 2 logical values of single bit, but also all superpositions of these 2 states. ψ = a 0 + b 1 Quantum state a and b are complex numbers which meets the equation: a 2 + b 2 = 1 Measuring a qubit will give us always 0 or 1, but at probabilities determined by a and b. Probability of measuring 0 is a 2 and probability measuring 1 is b 2 6

7 Update rules No measurement update rule When there is no measurement on the system, the state changes in a definite way that maintains any superposition (no randomness involved) Measurement update rule When there is a measurement on the system, the result determines the new state. The state is updated randomly depending on the outcome of the measurement. 7

8 Quantum entanglement Entangled objects cannot be fully described without considering the other(s). They remain in a quantum superposition and share a single quantum state. Measuring one particle determines value of the other particle. For example: We have total spin zero state (sum of all spins must be 0) with 2 particles. Before measurement none of the particles have definite spin, but by measuring one of them, the other instantly gains the opposite spin state. Particles need to interact with each other in order to get entangled. Ebit = Qbits of entanglement shared between communicating sides. Ebits itself cannot be used to send any information. 8

9 No cloning theorem Quantum state cannot be perfectly copied. Consequences We cannot use classical error correction (by using redundancy) during quantum computation, however it is possible to spread the information of one qubit onto a highly-entangled state of several (physical) qubits-> quantum error correction codes. no-teleportation theorem - quantum information cannot be measured with complete accuracy We cannot measure system in quantum state S, then prepare system according to the measurement. The prepared system will not be in the state S. Complete quantum information cannot be converted to classical information. 9

10 Bennets laws 1. qubit > bit 2. qubit > ebit = can do the job of 3. Dense coding qubit + ebit > 2 bits We can use 1 qubit and 1 ebit to send 2 bits of information. Only 1 qubit is transmitted between Alice and Bob. 4 possible rotations (no rotation, along X, Y, Z axis) = encoding 2 bits Bells measurement measures the type of entanglement of the spins can tell which rotation was made. Alice Bob 10

11 Bennets laws 4. Quantum teleportation ebit + 2 bits > qubit Transmission of quantum state by sending only 2 classical bits. Qubitoriginal state is destroyed during the Bells measurement Qubitin same quantum state as Alice had Qubit in uknown quantum state Alice Bob 11

12 Quantum cryptography If we distribute key any usual way using any algorithm it can be copied without us knowing, because classical information can be copied in principle (meaning no classical method of key distribution cannot be 100% secure). Quantum information cannot be copied in principle, thus it can be used to make unbreakable key distribution. 12

13 Quantum key distribution algorithm BB84 1. Alice sends Bob qubitswith random spin( )with equal likelihood. Noone can read whole sequence of the spins (because you can measure spins in Z axis or in X axis, but not both X and Z are complementary variables). 2. Bob randomly measures X or Z axis. 3. Bob publicly sends Alice which axis he measured. 4. Alice publicly replies which measurements are good and which are bad (good measurement = measuring the appropriate axis). 5. Alice and Bob discards the bad measurements and the good measurements become the key. 6. Alice and Bob publicly compare some bits of their key (which are then not used) and if they do not match, it tells them that someone measured their key during step 1) and key is discarded and will not be used. 13

14 BB84 - Example Alice sends: Bob measures: Z Z X Z X X Z Bob s results: Bob sends: Z Z X Z X X Z Alice sends: Key: Real quantum states of qubits stays private with Alice. Measurements stays private with Bob. 14

15 Quantum computers Computer where memory elements are qubits. Qubits can be any particle that has 2 states that are distinguishable. During a computation, quantum computer operates without any measurement -> computer must be well isolated from outside world (any interaction with surroundings counts as a measurement). Quantum register of 14 calcium atoms 15

16 Quantum computers Computer follows multiple computational paths at the same time (same as photon can follow more than one path in interferometer). During computer operation no physical record is made which of these computation is done. At the end it combines all results of computational paths and using interference we get our measurement. 16

17 Quantum algorithms In principle quantum computer cannot solve anything that classical computer can t, but there are algorithms that use quantum mechanics to run computation asymptoticaly faster. Deutsch Jozsaproblem First problem that showed quantum computers can be more powerful than ordinary computers. We have a function that takes number from 1-M. Output of this function is either constant (same 0 or 1) or balanced (returns 1 for half inputs and 0 for other half). Classical algorithm has to examine M/2 +1 function values in order to determine if its constant or balanced. Quantum computer can calculate that by calculating the function only once using superposition of all possible inputs. In this process we don t know anything about individual values of the function, but we find out property of all the values taken together. 17

18 Deutsch Jozsa problem Another way to explain how this works is via two slit experiment involving electrons, with a electromagnet in the middle. We now wish to determine if the electromagnet is turned on or off. This can be determined by looking at the interference pattern without any need of measuring the individual electrons -> superposition state is like interference pattern. Electromagnet 18

19 Quantum algorithms Integer factorizationhas exponential complexity and you cannot have faster algorithm on standard computer. But using Shor s algorithm we can solve the same problem on quantum computer with polynomial complexity log 10 (time) General number field sieve Lots of modern cryptography is based on difficulty of factoring numbers Shor s algorithm 10 2 Length of factored integer [bits] 19

20 Quantum algorithms Shor s algorithm The first part of the algorithm turns the factoring problem into the problem of finding the period of a function, and may be implemented classically. The second part finds the period using the quantum Fourier transform, and is responsible for the quantum speedup. The speedup is hidden within quantum superposition and ability to evaluate function at all points simultaneously. Grover's algorithm Algorithm searches unsorted database with N entries in O(N 1/2 ) time and using log(n) storage space. Classical algorithm must search every item -> linear complexity. 20