Quantum Mechanics 103

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Quantum Mechanics 103

• Quantum Implications for Computing

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Schrödinger and Uncertainty

• Going back to Taylors experiment, we see that
  the wavefunction of the photon extends through
  both slits
• Therefore the photon has traveled through both
  openings simultaneously
• The wavefunction of a particle will contain
  every possible path the particle could take until
  the particle is detected by scattering or being
  absorbed
• These paths can interfere with each other to
  produce diffraction-like probability patterns
• BUT, Schrödinger took this explanation to an
  extreme

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Schrödingers Famous Cat

• Suppose a radioactive substance is put in a box
  with a cat for a period of time

• If the Geiger Counter triggers, a gun is
  discharged and the cat is killed

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Schrödingers Famous Cat

• Until an observer opens the box to make a
  measurement of the system,

• The nucleus remains both decayed

and undecayed

• The Geiger counter remains both triggered

and untriggered

• The gun has both fired

and not fired

• The cat is both dead

and alive
Disclaimer To be truly indeterministic, this
experiment must be performed in a sound-proof
room with no window
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Paradox?

• Paradoxical as it may seem, the concept of
  superposition of states is borne out well in
  experiment
• Like superposition of waves producing
  interference effects
• Quantum Mechanics is one of the most-tested and
  best-verified theories of all time
• But it seems counter-intuitive since we live in a
  macroscopic world where uncertainty on the order
  of ? is not noticeable

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Quantum paradox 2

• Einstein-Podolsky-Rosen (EPR) paradox
• Consider two electrons emitted from a system at
  rest measurements must yield opposite spins if
  spin of the system does not change
• We say that the electrons exist in an entangled
  state

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More EPR

• If measurement is not done, can have interference
  effect since each electron is superposition of
  both spin possibilities
• But, measuring spin of one electron destroys
  interference effects for both it and the other
  electron
• It also determines the spin of the other electron
• How does second electron know what its spin is
  and even that the spin has been determined

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Interpreting EPR

• Measuring one electron affects the other
  electron!
• For the other electron to know about the
  measurement, a signal must be sent faster than
  the speed of light!
• Such an effect has been experimentally verified,
  but it is still a topic of much debate

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Interference effects

• Remember this Mach-Zender Interferometer?
• Can adjust paths so that light is split evenly
  between top U detector and lower D detector, all
  reaches U, or all reaches D due to interference
  effects
• Placing a detector (either bomb or
  non-destructive) on one of the paths means 50
  goes to each detector ALL THE TIME

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Interpretation

• Wave theory does not explain why bomb detonates
  half the time
• Particle probability theory does not explain why
  changing position of mirrors affects detection
• Neither explains why presence of bomb destroys
  interference
• Quantum theory explains both!
• Amplitudes, not probabilities add - interference
• Measurement yields probability, not amplitude -
  bomb detonates half the time
• Once path determined, wavefunction reflects only
  that possibility - presence of bomb destroys
  interference

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Quantum Theory meets Bomb

• Four possible paths RR and TT hit upper
  detector, TR and RT hit lower detector
  (Rreflected, Ttransmitted)
• Classically, 4 equally-likely paths, so prob of
  each is 1/4, so prob at each detector is 1/4
  1/4 ½, independent of path length difference
• Quantum mechanically, square of amplitudes must
  each be 1/4 (prob for particular path), but
  amplitudes can be imaginary or complex!
• This allows interference effects

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What wave function would give 50 at each
detector?

• Must have a2 b2 c2 d2 1/4
• Need a b2 cd2 1/2

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If Path Lengths Differ, Might Have

• Lower detector
• Upper detector

Voila, Interference!
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When Measure Which Path,

• Lower detector
• Upper detector

Voila, No Interference!
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Quantum Storage

• Consider a quantum dot capacitor, with sides 1 nm
  in length and 0.010 microns between plates
• How much energy required to place a single
  electron on those plates?
• Can make confinement of dot dependent upon
  voltage
• Lower the voltage, let an electron on gt 1
• Lower voltage on other side, let the electron off
  -gt 0

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What must a computer do?

• Deterministic Turing Machine still good model
• Two pieces
• Read/write head in some internal state
• Infinite tape with series of 1s, 0s, or blanks
• Follows algorithms by performing 3 steps
• Read value of tape at heads location
• Write some value based on internal state and
  value read
• Move to next value on tape

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Can we improve this model?

• Probabilistic Turing Machine sometimes better
• Multiple choices for internal state change
• Not 100 accurate, but accuracy increases with
  number of steps
• Can solve some types of problems to sufficient
  accuracy much more quickly than deterministic TM
  can
• Similar concept to Monte Carlo integration

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Limits on Turing Machines

• Some problems are solvable in theory but take too
  long in practice
• e.g., factoring large numbers
• Can label problems by how the number of steps to
  compute grows as the size of the numbers used
  grows
• addition grows linearly
• multiplication grows as the square of digits
• Fourier transform grows faster than square
• factoring grows almost exponentially

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Examples of factoring time

• MIP-year 1 year of 1 million processes per
  second
• Factoring 20-digit decimal number done in 1964,
  requiring only 0.000009 MIP-years
• 45-digit decimal number (1974) needs 0.001
  MIP-years
• 71-digit decimal number (1984) needs 0.1
  MIP-years
• 129-digit decimal number (1994) needs 5000
  MIP-years

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Quantum Cryptography

• Current best encryption uses public key for
  encoding
• Need private key (factors of large integer in
  public key) to decode
• Really safe unless
• Someone can access your private key
• Quantum computers become prevalent

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Quantum Cryptography II

• Quantum Computers can factor large numbers
  near-instantly, making public key encryption
  passe
• But, can send quantum information and know
  whether it has been intercepted

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What problems face QC?

• Decoherence if measurement made, superposition
  collapses
• Even if measurement not intentional!
• i.e., if box moves, cat becomes alive or dead,
  not both
• Quantum error correction
• No trail of path taken (or else no superposition)
• Proven to be possible that doesnt mean its
  easy!
• HUGE Technical challenges
• electronic states in ion traps (slow, leakage)
• photons in cavity (spontaneous emission)
• nuclear spins in molecule (small signal in large
  noise)