Quantum Computing. Part I. Thorsten Altenkirch
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1 Quantum Computing Part I Thorsten Altenkirch
2 Is Computation universal? Alonzo Church - calculus Alan Turing Turing machines computable functions
3 The Church-Turing thesis All computational formalisms define the same set of computable functions
4 The Church-Turing thesis All computational formalisms define the same set of computable functions What do we mean by all formalisms?
5 The Church-Turing thesis (CTT) All physically realizable computational formalisms define the same set of computable functions Most people believe CTT Hypercomputing?
6 The Church-Turing thesis All physically realizable computational formalisms define the same set of computable functions Most people believe CT Hypercomputing? Throw you TM in a black hole...
7 Feasible computation? Not all computable problems can be solved in practice. TAUT Example: Is (P Q R) (P Q R) a tautology? The best known algorithm for TAUT requires exponential time in the number of propositional variables.
8 The extended Church-Turing thesis (ECT) All physically realizable computational formalisms define the same set of feasible computable functions Challenged by non-standard computational formalisms P-systems inspired by biology Quantum Computing inspired by quantum physics
9 Factor & Primes FACTOR Input: a number (e.g. 15) Output: a (nontrivial) factor (e.g. 3 or 5) or prime PRIMES Input: a number (e.g. 15, 7) Output:yes (e.g. for 7) no (e.g. for 15) The best known algorithm for factorisation needs exponential time. Hence factorisation is not feasible. However, there is a polynomial algorithm for PRIMES (feasible). Important for public key cryptography (e.g. RSA)
10 Shor s algorithm Peter Shor (MIT) 1994 : Shor develops an (probabilistic) algorithm that would solve FACTOR in polynomial time on a (hypothetical) quantum computer. This indicates that the ECT doesn t hold for quantum computing
11 Activities Implement a program for factorisation in your favourite programming language. Try to make it as efficent as possible. Analyze its time complexity. Implement an efficient primality checker using a probabilistic algorithm (Hint: lookup Primality test on wikipedia).
12 Quantum Physics for dummies
13 Is light wave or particle? Light is made of particles Light is a wave Isaac Newton ( ) Christiaan Huygens ( )
14 Young s double slit experiment In 1801 Thomas Young performed the double slit experiment It produces an interference pattern Light is a wave!
15 The photoelectric effect Photons can dislocate electrons in certain materials The energy of the electrons only depends on the frequency of the light Not on intensity! Below a certain intensity no electrons can dislocated. Conclusion: light consists of particles (photons).
16 Wave particle duality Electrons also behave this way. Copenhagen interpretation Particles are probability waves Amplitude corresponds to the probability that we observe the particle. Niels Bohr ( )
17 Einstein-Podelsky-Rosen paradox (EPR) Thought experiment about non-locality Two particles with opposite spin are produced. They are measured far apart. According to QM we always measure both particle if we measure one. EPR argue that there have to be hidden variables
18 Bell inequality Local variables or non-locality? Is there an experiment to show who is right? Bell showed that there are experiments which refute hidden-variable theory An intuitive account of Bell s theorem has been given by Mermin
19 Mermin s thought experiment Each time we press the button on C both detectors show a red or green light. If both detectors have the same settings (1,2,3) the same light goes on. If the detectors have different settings the same light goes on in 1/4 of the cases.
20 The conundrum of the device Each particle has to carry the information which light to flash for each setting (3 bits). Both particles have the same instruction set. Assume the instruction set is RRG. If we measure different bits, the same light goes on in at least 1/3 of all cases. The same is true for all other instruction sets. This is incompatible with the observation that same light went on only in 1/4 of all cases!
21 Non-locality rules! Mermin s experiment cannot explain by hidden variables. QM has a simple explanation: We are measuring the spin of entangled particles with orientations 0º, 120º, 240º. If the settings are different, the probability that the measurements agree is given by and: cos =1 cos = cos = 1 4 cos 2 θ 2 While Mermin s experiment is a thought experiment, similar experiments have been carried out in practice.
22 Activities Compile a survey about experiments to test the Bell-inequality. What are the issues? Read Mermin s article (on the webpage, only 5 pages, easy reading) and try to explain it to sb else. Record your conversation. Implement a program (in your favourite language) that simulates Mermin s experiment using the quantum mechanic explanation (i.e. using cos and random numbers).
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