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Published on October 16, 2007

Author: Lucianna

Source: authorstream.com

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Simulating Physics with Computers :  Simulating Physics with Computers Richard Feynman, 1982 Michael Demmer, Rodrigo Fonseca, Farinaz Koushanfar UC Berkeley, Fall 2004 Richard Feynman:  Richard Feynman Was born on May 11, 1918, in Brooklin. Moved to Far Rockaway, New York, at 10. His father Melville Feynman Was influential in his career and formed the essence of Feynman’s way of understanding Taught him to question things around him and to try to find explanations I was born not knowing and have had only a little time to change that here and there. ~ Richard Feynman Early Portraits:  Early Portraits Pre-War:  Pre-War Met Arline Greenbaum in high school Attended MIT (1935-1939) Moved to Princeton for his PhD in 1939 Proposed to Arline in Princeton, planned marriage after PhD Arline was positively diagnosed with tuberculosis, they got married immediately US entered World War in December 1941 Young Days:  Young Days "(...)the idea seemed so obvious to me and so elegant that I fell deeply in love with it. And, like falling in love with a woman, it is only possible if you do not know much about her, so you cannot see her faults”. ~ Feynman, about the idea that led to his Nobel prize Manhattan Project:  Manhattan Project His PhD @ Princeton: the probability of a transition of a quantum from one state to some subsequent state Entirely new formalism in quantum mechanics, adapted it to the physics of QED For this, he was awarded the Nobel Prize in physics, shared with Schwinger and Tomonaga (1965) Moved to Los Alamos, NM, in 1942 to work on the Manhattan project In July of 1945, Arline passed away He is by all odds the most brilliant young physicist here [at Los Alamos], and everyone knows this. ~ J. Robert Oppenheimer Professorship:  Immediately accepted a job at Cornell Moved to Caltech in 1950, married 2nd wife In the early 1960s, was assigned the lectures in physics that took him 3 years In 1960, married to 3rd wife, Gweneth In 1965, Feynman received the Nobel Prize for his work in QED Professorship If I could explain it to the average person, I wouldn't have been worth the Nobel Prize. ~ Richard Feynman At Caltech…:  There are two types of genius. Ordinary geniuses do great things, but they leave you room to believe that you could do the same if only you worked hard enough. Then there are magicians, and you can have no idea how they do it. Feynman was a magician. ~ Hans Bethe At Caltech… More Richard Feynman..:  More Richard Feynman.. Made a breakthrough in the physics of the superfluidity of super cold liquid helium Helium shows quantum mechanical behavior at macroscopic scales Worked on "weak decay", in the decay of a free neutron into an electron, a proton, and an anti-neutrino w/ Murray Gell-Mann Shared the results w Marshak and Sudarshan Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. ~ Richard Feynman Late Richard Feynman:  Late Richard Feynman In 1979, he was diagnosed with a rare form of cancer growing in his abdomen In 1980s, Feynman became very popular "Surely You're Joking, Mr. Feynman!“ "What Do You Care What Other People Think?" both published by Ralph Leighton Investigated Challenger accident in 1986 Feynman passed away on Feb. 15, 1988 I would hate to die twice. It’s so boring. ~ Feynman’s last words A curious character...:  A curious character... “What Happened to Tanna Tuva?”:  “What Happened to Tanna Tuva?” As a boy, Feynman collected stamps from Tuva Tuva!? Ralph Leighton, Friends of Tuva Kyzyl was a center for nuclear research Interesting culture Center of Asia Famous throat singers Feynman is a hero in Tuva Ended up never going there! Brazil:  Brazil He spent two periods in Brazil, teaching physics at a university in Rio Learned portuguese Learned to play samba, and was part of a local samba club! Drums:  Drums In 1966 a Swedish encyclopedia publisher asked for a picture of Feynman "beating the drum" to give "a human approach to a presentation of the difficult matter that theoretical physics represents". Feyman’s reply: Dear Sir, The fact that I beat a drum has nothing to do with the fact that I do theoretical physics. Theoretical physics is a human endeavor, one of the higher developments of human beings, and the perpetual desire to prove that people who do it are human by showing that they do other things that a few other humans do (like playing bongo drums) is insulting to me. I am human enough to tell you to go to hell. Yours, RPF Los Alamos:  Los Alamos Motivation “The Germans had Hitler and the possibility of developing an atomic bomb was obvious, and the possibility that they would develop it before we did was very much of a fright”. Supervised “computers” “The only difference is that the IBM machines didn’t get tired and could work three shifts. But the girls got tired after a while” Lock picking “I was always dumb in that way. I never knew who I was talking to. I was always worried about the physics. If the idea looked lousy, I said it looked lousy.” ~ After meeting Niels Bohr Feynman’s Van:  Feynman’s Van 1975 Dodge Tradesman Maxivan, bought new and outfitted in Long Beach Had Feynman’s diagrams painted Sold for $1 to Leighton, who used it to transport visiting Tuvan throat singers! Challenger:  “Dr. Feynman was, in my opinion, the most personally and professionally objective member and I might add the ONLY fearless member concerning potential career damage”. Roger M. Boisjoly, M.Thiokol Engineer Feynman went directly to the engineers, and found out the O ring which was the culprit for the explosion. Challenger “For a successful technology, reality must take precedence over public relations, for nature cannot be fooled”. ~ Final words of the Challenger report There’s much more...:  There’s much more... Mayan hieroglyphs Drawing Advice on getting women at bars... His books are a great read... Feynman on Quantum Mechanics:  Feynman on Quantum Mechanics “(secret, secret, close the doors!) we have always had a great deal of difficult in understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t gotten to the point that this stuff is obvious to me. Okay, I still get nervous with it.” Quantum Effects:  Quantum Effects A weak light source is set up to point at a sensitive detector that ‘clicks’ when individual photons are detected Light acts like a particle: dimmer light reduces frequency not amplitude of detections But other experiments (e.g. double slit interference) show that light behaves like a wave light source detector 2 Quantum Effects (2):  Quantum Effects (2) light source detector 1 detector 2 half-silvered mirror When a half-silvered mirror is placed in the path, ½ of the photons pass through the mirror and ½ are reflected. Therefore photons are detected at each location with equal probability But how does it “know” which way to go? Newton had a hard time explaining this And where is the photon immediately after passing through the mirror? Quantum Effects (3):  Quantum Effects (3) light source detector 1 detector 2 half-silvered mirror full mirror Now force the split beams back together, then send through another half-silvered mirror Classical mechanics would predict that again 50% would be detected at each location Instead all the photons are detected at one location! Somehow it “knows” that it shouldn’t go to detector 2 Are some photons are pre-disposed to reflect, and others to pass through the mirror? Or does each photon actually go both ways at the same time… Quantum Effects (4):  Quantum Effects (4) light source detector 1 detector 2 half-silvered mirror full mirror When one path is blocked, then strange things really start… The probability is again evenly split among the two detectors The photon must take both paths at the same time (or go back in time) Once it passes through the first mirror, each photon is in a coherent superposition of the two states The state is only fully determined when it is measured, which destroys the superposition and forces it one way or the other From Bits to Qubits:  From Bits to Qubits In a quantum computer, a superposition is used as the fundamental unit of data, called a qubit e.g. an atom, or nuclear spin, or a polarized photon When measured, a qubit is in only one of two states Represented in Dirac notation as a ket: for example the state of a spin ½ particle is measured as |+½ (spin up) or |-½ (spin down) Can be used as digits, assigning one spin to 0 and the other to 1 But until it’s measured, a qubit is actually in a combination of state 0 and state 1 The probability distribution cannot be measured directly But, it can be used in computation… From Bits to Qubits (2):  From Bits to Qubits (2) A bit of mathematical formalism: A qubit is a unit state vector in a two dimensional Hilbert space where |0 and |1 are orthonormal basis vectors For each qubit |x there exist two (complex) numbers a, b s.t. |x = a|0 + b|1 and |a|2 + |b|2 = 1 So a and b define the angle which the qubit makes with the vertical axis and therefore the probability that the given bit will be measured as a 0 or as a 1 There’s also the phase which represents an angle of rotation around the vertical axis Doesn’t affect the value of the bit, but is crucial for quantum interference effects Qubit evolution:  Qubit evolution Similar to a classical register, register of 3 physical qubits can store 23 = 8 values Of course, these values are in a superposition So in effect, the register stores all 8 values at once, with a probability distribution on the set of values Still, a qubit contains no more information than a classical bit The reason is that once you measure the value, it is forced into one of the two states The quantum analog to a classical operator is an evolution Transforms an input by some process to an output register E.g. rotation: |0  cosΘ|0 + sinΘ|1, |1  -sinΘ|0 + cosΘ|1 Evolutions operate without measuring the value of a qubit Thus it creates a new superposition Essentially performs a parallel computation on all the values at once Measurement and Entanglement:  Measurement and Entanglement Quantum states cannot be cloned Measuring forces a superposition it into state 0 or state 1 Seems “bad” for most general computing purposes But is pretty useful if you’re trying to communicate a secret key… Measuring one bit can affect another Consider a two bit system: (1/2) (|00 + |11) Although the probability that the first bit is |0 is 1/2, once the second bit is measured, then this probability is either 0 or 1! This is called entanglement Not all states are entangled, e.g. (1/2)(|00 + |01) Measuring can even kill the cat Shrödinger described: |cat = (1/2) (|dead + |alive) Error control codes:  Error control codes Turing machines Classical computers are based around assumptions (rightly) that values can be measured and manipulated reliably Though implementations may require energy input to maintain state, theoretically irrelevant to the computations Shannon and Information Theory Principles of error correction over a communication channel lead to a new field Still, the applications are constrained to multi-party communications, not related to internal mechanics of a computer Quantum Computers Quantum computations turn out to be very sensitive to noise in the environment A natural fit for error correction codes Thus a deeper relationship is likely to exist between Information Theory and Quantum Computing than in the classical case (Pre)History of Quantum Computing:  (Pre)History of Quantum Computing Thermodynamics and Computation 1871: Maxwell’s Demon 1929: Szilard reduces the problem to particle identification (and introduces the concept of a “bit” of information but not the term) 1961: Landauer shows that erasure of information is dissipative and therefore irreversable 1970s: Bennett, Fredkin, Toffoli, etc. apply these ideas to general computation 1973: Bennett, shows that any computation is reversible, i.e. no entropy cost (e.g. Toffoli replacement for a NAND gate) 1982: Bennett applies to Maxwell’s demon showing it requires energy to erase its memory (Pre)History of Quantum Computing:  (Pre)History of Quantum Computing Quantum links to Information Theory 1935: Einstein, Podolsky, Rosen describe gedanken experiment in which quantum experiments suggest effects at a distance, claim it to be a hole in the theory “God does not play dice with the universe” 1964: Bell analyzes EPR conundrum and proposes that no hidden variable theory can reproduce quantum theory predictions – therefore nonlocal interactions can exist 1982: Aspect, Dalibard, Roger support Bell’s theorem showing that any interaction must travel faster than the speed of light Quantum Mechanics / Information Theory:  Quantum Mechanics / Information Theory \ History of Quantum Computing:  History of Quantum Computing 1980: Benioff describes a hybrid Turing machine that stores qubits on the tape 1982: Feynman considers simulation of quantum systems by a quantum computer 1984: Albert describes a 'self measuring quantum automaton' that performs tasks no classical computer can simulate 1982-4: Weisner, Bennett examine quantum key exchange 1985: Deutsch specifies and describes a universal quantum computer 1993: Simon describes oracle problem for which quantum computers are exponentially faster than classical ones 1994: Shor describes quantum algorithm for efficient factorization of large numbers 1995: Shor proposes quantum error correction 1997: Bernstein, Vazirani on quantum complexity theory 1998: First working 2-qbit NMR computer at UCB 2001: 7-qubit NMR computer at IBM Almaden executes Shor’s algorithm to factor the number 15 Possibilities in Computer and Possibilities in Physics:  Possibilities in Computer and Possibilities in Physics Can quantum physics be simulated by a universal computer? Modifying the physical laws may cause anisotropies Early conception: natural laws are reversible but physical laws are not! Computer reversibility: Bennet, Fredkin Toffoli Possibilities in computers and possibilities in Physics! Science is the belief in the ignorance of experts. ~ Richard Feynman Simulating Time:  Simulating Time Rule of simulation: Number of computer elements must be proportional to the space-time volume of the physical system For simulation, assume time is discrete Simulating time in cellular automata: The computer is going from a state to a state It is not simulated! It is imitated! Is there a way to simulate rather than imitate? Space-Time Example:  Space-Time Example State si is a function of states m,k in its neighborhood, Si = Fi (sm, sk, ….) What if F depends on both future and the past? Suppose that you now Fi, that is a function of future vars… How to choose numbers to satisfy equations? Classical physics is local, causal and reversible… Space Time Si Sm Sk Simulating Probability:  Simulating Probability We have difficulty in understanding quantum mechanical view of the world! One way to simulate a probabilistic theory is to calculate the probability and interpret this number to represent nature! Problem with discretizing probability. If we have R particles, we need k-digits for every configuration x1, …,xR at time t. For N space points  NR! Exponential!!! IMPOSSIBLE! Probabilistic Computer:  Probabilistic Computer Simulate the probabilistic nature by a probabilistic computer Imitating, but… nature is unpredictable: Take a Monte Carlo simulation approach! Local probabilistic computer: Determine the behavior in one region by disregarding the events in other regions! Probability of Transition:  Probability of Transition If each point i=1,…,N in space has state si, w/ probability P{si}, at each time: Pt+1({s})= [ i m(si|s’k,s’h,…)] Pi({s’}) As k moves far from i, m becomes less sensitive to s’k Probability of making a transition The same as cellular automata, instead of being definite, it’s a probability How to simulate quantum mechanical effects?:  How to simulate quantum mechanical effects? For a single particle,  is a function of x and t and we can use a probabilistic eq. Full description of quantum mechanics for a large system w/ R particles cannot be simulated in polynomial time in R or N! There are two ways to go around this: Let the computer itself be built by quantum mechanical elements that obey quantum rules Can we imitate this on a universal computer? Quantum simulators:  Quantum simulators He proposes the idea of a quantum computer, different from a Turing machine You could imitate any quantum system Leaves open: to work out classes of intersimulatable quantum systems Polarization of Photons:  Polarization of Photons if you're doing an experiment, you should report everything that you think might make it invalid — not only what you think is right about it... Two state systems:  Two state systems Each photon either goes to the O or E detectors Only one detector P(O) + P(E) = 1 Two state systems:  Two state systems For each photon, only one detector is triggered P(O|O) = cos2; P(E|O) = 1 - cos2 = sin2 P(E|E) = sin2; P(O|E) = 1 - sin2 = cos2 All right so far… Two photon correlation:  Two photon correlation One atom emits two photons simultaneously Two detectors at 1 and 2 By Quantum theory and experiment POO = PEE = ½ cos2(2 - 1) POE = PEO = ½ sin2(2 - 1) You can always predict what I get: set 2 = 1  POE = PEO = 0 Do not keep saying to yourself, if you can possible avoid it, "But how can it [Quantum behaviour] be like that?" because you will get "down the drain," into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. Two photon correlation:  Two photon correlation It turns out you can’t simulate this on a local probabilistic computer ... squeeze into a numerical question ...:  ... squeeze into a numerical question ... Suppose 2 - 1=30º, what’s the probability that get the same result? In this case, it’s 2/3 For all possible 8 configurations, it’s <= 2/3 But quantum mechanics, and experiment, yield cos2(30º) = ¾ ! So...:  So... “This kind of logic” cannot reproduce this result Things could be affected by the future as well Instantaneous communication (non-local) Origin of quantum probabilities: maybe we are correlated with any experiment we do “(...) you people who think about computer-simulation possibilities (...) see if you can’t invent a different point of view than the physicists have had to invent (...)” Thinking of computation has led to progress in other areas Future of Quantum Computing:  Future of Quantum Computing (according to Christos) Someone will build a functional quantum computer After years of repeated roadblocks and failed efforts, the field will fizzle out and die Continued work into QC will lead to a fundamental change in the understanding of quantum mechanics itself. “…and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy. Thank you.”

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