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					Simulating Physics with Computers
          Richard Feynman, 1982
  Michael Demmer, Rodrigo Fonseca, Farinaz Koushanfar
               UC Berkeley, Fall 2004
   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
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




   "(...)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

    • 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




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
      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

• 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...
 “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

• 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
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

       • 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

• 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.




“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...

• Mayan hieroglyphs
• Drawing
• Advice on getting women at bars...

• His books are a great read...
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

light source




                                                      detector 2




   • 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
Quantum Effects (2)
                                                                          detector 1

light source



                           half-silvered mirror




                                                                           detector 2




   • 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)

light source                                                               detector 1
                                                  full mirror


                           half-silvered mirror



                                                                           detector 2




   • 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)

light source                                                               detector 1
                                                  full mirror


                           half-silvered mirror



                                                                           detector 2




   • 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

• 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)

• 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

• 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

• 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

• 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

• 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

• 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




                  \
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
• 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

• 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

                      Sm

Time             Si
                      Sk
                           Space
• 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…
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.
       IMPOSSIBLE!
• If we have R particles, we need k-digits for
  every configuration x1, …,xR at time t.
• For N space points  NR! Exponential!!!
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

• 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?
• 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

• 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




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




• Each photon either goes to the O or E
  detectors
  – Only one detector
  – P(O) + P(E) = 1
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

        • 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

• It turns out you can‟t simulate this on a
  local probabilistic computer
... 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...

• “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

  (according to Christos)

  1.    Someone will build a functional quantum computer

  2.    After years of repeated roadblocks and failed
        efforts, the field will fizzle out and die

  3.    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.”